{"id":4992,"date":"2023-12-21T11:12:59","date_gmt":"2023-12-21T05:42:59","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=4992"},"modified":"2023-12-22T14:56:27","modified_gmt":"2023-12-22T09:26:27","slug":"envision-math-common-core-grade-8-answer-key-topic-5","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-8-answer-key-topic-5\/","title":{"rendered":"enVision Math Common Core Grade 8 Answer Key Topic 5 Analyze And Solve Systems Of Linear Equations"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 8 Answer Key<\/a>\u00a0Topic 5 Analyze And Solve Systems Of Linear Equations<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 8th Grade Answers Key Topic 5 Analyze And Solve Systems Of Linear Equations<\/h2>\n

Topic Essential Question<\/strong>
\nWhat does it mean to solve a system of linear equations?
\nAnswer:
\nA system of linear equations is just a set of two or more linear equations. In two variables (x and y), the graph of a system of two equations is a pair of lines in the plane<\/p>\n

Topic 5 \u00ebnVision STEM Project<\/strong><\/p>\n

Did You Know?
\nAfter the Boston Tea Party of 1773, many Americans switched to drinking coffee rather than tea because drinking tea was considered unpatriotic.
\n\"Envision
\nAlthough Brazil is the largest coffee-producing nation in the world, Americans combine to drink 0.2% more coffee each year than Brazilians. The third-ranked nation for total coffee consumption, Germany, consumes approximately 44% as much coffee as either the United States or Brazil.<\/p>\n

The United States consumes the most coffee by total weight, but Americans do not drink the most coffee per capita. People in northern European countries like Finland, Norway, and Holland drink more than twice as much coffee as their American counterparts each day.
\n\"Envision
\nIn some coffee-producing nations, millions of acres of forest are cleared to make space for coffee farming. Sustainable farms grow coffee plants in natural growing conditions without chemicals and with minimal waste.
\n\"Envision
\nCoffee beans are actually seeds that are harvested from cherries that grow on coffee plants in tropical climates.<\/p>\n

Your Task: Daily Grind
\n\"Envision
\nCoffee roasters create coffee blends by mixing specialty coffees with less expensive coffees in order to create unique coffees, reduce costs, and provide customers with consistent flavor. You and your classmates will explore coffee blends while considering the environmental and economic impact of the coffee trade.<\/p>\n

Topic 6 GET READY!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary<\/strong>
\nChoose the best term from the box to complete each definition.
\nlinear equation
\nparallel
\nslope
\ny-intercept<\/p>\n

Question 1.
\nThe value of m in the equation y = mx + b represents the __________ .
\nAnswer:
\nWe know that,
\nThe value of m in the equation
\ny = mx + b
\nrepresents the “Slope”
\nHence, from the above,
\nWe can conclude that the best term from the box to complete the given definition is “Slope”<\/p>\n

\"Analyze<\/p>\n

Question 2.
\nWhen lines are the same distance apart over their entire lengths, they are _________.
\nAnswer:
\nWe know that,
\nWhen lines are the same distance apart over their entire lengths, they are “Parallel”
\nHence, from the above,
\nWe can conclude that the best term from the box to complete the given definition is “Parallel”<\/p>\n

Question 3.
\nThe _________ is the value b in the equation y = mx + b.
\nAnswer:
\nWe know that,
\nThe “y-intercept” is the value of b in the equation
\ny = mx + b
\nHence, from the above,
\nWe can conclude that the best term from the box to complete the given definition is “y-intercept”<\/p>\n

Question 4.
\nA __________ is a relationship between two variables that gives a straight line when graphed.
\nAnswer:
\nWe know that,
\nA “Linear relationship” is a relationship between two variables that gives a straight line when graphed
\nHence, from the above,
\nWe can conclude that the best term from the box to complete the given definition is “Linear relationship”<\/p>\n

Identifying Slope and y-Intercept<\/strong><\/p>\n

Identify the slope and the y-intercept of the equation.
\nQuestion 5.
\ny = 2x – 3
\nslope = _________
\ny-intercept = ________
\nAnswer:
\nThe given equation is:
\ny = 2x – 3
\nCompare the above equation with
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the y-intercept
\nHence, from the above,
\nWe can conclude that
\nThe slope is: 2
\nThe y-intercept is: -3<\/p>\n

Question 6.
\ny =-0.5x + 2.5
\nslope = _________
\ny-intercept = ________
\nAnswer:
\nThe given equation is:
\ny = -0.5x + 2.5
\nCompare the above equation with
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the y-intercept
\nHence, from the above,
\nWe can conclude that
\nThe slope is: -0.5
\nThe y-intercept is: 2.5<\/p>\n

Question 7.
\ny – 1 = -x
\nslope = _________
\ny-intercept = ________
\nAnswer:
\nThe given equation is:
\ny – 1 = -x
\nAdd 1 on both sides
\nSo,
\ny = -x + 1
\nNow,
\nCompare the above equation with
\ny = mx + b
\nWhere,
\nm is the slope
\nb is the y-intercept
\nHence, from the above,
\nWe can conclude that
\nThe slope is: -1
\nThe y-intercept is: 1<\/p>\n

Graphing Linear Equations<\/strong><\/p>\n

Graph the equation.
\nQuestion 8.
\ny = \\(\\frac{2}{3}\\)x – 2
\n\"Envision
\nAnswer:
\nThe given equation is:
\ny = \\(\\frac{2}{3}\\)x – 2
\nHence,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"<\/p>\n

Question 9.
\ny = -2x + 1
\n\"Envision
\nAnswer:
\nThe given equation is:
\ny = -2x + 1
\nHence,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"<\/p>\n

Solving Equations for Variables<\/strong><\/p>\n

Solve the equation for y.
\nQuestion 10.
\ny – x = 5
\nAnswer:
\nThe given equation is:
\ny – x = 5
\nAdd x on both sides
\nSo,
\ny – x + x = 5 + x
\ny = x + 5
\nHence, from the above,
\nWe can conclude that the value of y is:
\ny = x + 5<\/p>\n

\"Analyze<\/p>\n

Question 11.
\ny + 0.2x = -4
\nAnswer:
\nThe given equation is:
\ny + 0.2x = -4
\nSubtract with 0.2x on both sides
\nSo,
\ny + 0.2x – 0.2x = -4 – 0.2x
\ny = -0.2x – 4
\nHence, from the above,
\nWe can conclude that the value of y is:
\ny = -0.2x – 4<\/p>\n

Question 12.
\n–\\(\\frac{2}{3}\\)x + y = 8
\nAnswer:
\nThe given equation is:
\n–\\(\\frac{2}{3}\\)x + y = 8
\nAdd with \\(\\frac{2}{3}\\)x on both sides
\nSo,
\n–\\(\\frac{2}{3}\\)x + \\(\\frac{2}{3}\\)x + y = 8 + \\(\\frac{2}{3}\\)x
\ny = \\(\\frac{2}{3}\\)x + 8
\nHence, from the above,
\nWe can conclude that the value of y is:
\ny = \\(\\frac{2}{3}\\)x + 8<\/p>\n

Language Development<\/strong><\/p>\n

Complete the fishbone map by writing key terms or phrases related to systems of linear equations on each diagonal. Connect supporting ideas on the horizontal lines.
\n\"Envision<\/p>\n

Topic 5 PICK A PROJECT<\/h3>\n

PROJECT 5A
\nWhat is the most interesting discussion you’ve had with your classmates?
\nPROJECT: WRITE A SPEECH FOR A DEBATE
\n\"Envision<\/p>\n

PROJECT 5B
\nWhat can you compare with a Venn diagram?
\nPROJECT: DRAW A VENN DIAGRAM
\n\"Envision<\/p>\n

PROJECT 5C
\nWhat can you do with a smartphone?
\nPROJECT: CHOOSE A CELL PHONE PLAN
\n\"Envision<\/p>\n

PROJECT 5D
\nIf you made a stained-glass window, what colors and shapes would you use?
\nPROJECT: MAKE A MODEL OF A STAINED-GLASS WINDOW
\n\"Envision<\/p>\n

Lesson 5.1 Estimate Solutions by Inspection<\/h3>\n

Solve & Discuss It!<\/strong>
\nDraw three pairs of lines, each showing a different way that two lines can intersect or not intersect. How are these pairs of lines related?
\n\"Envision
\nI can… find the number of solutions of a system of equations by inspecting the equations.
\nAnswer:
\nThe representation of three pairs of lines, each showing a different way that two lines can intersect or not intersect is:
\n\"\"
\nNow,
\nParallel lines:
\nThe lines that do not intersect each othse and have the same slope but different y-intercepts
\nPerpendicular lines:
\nThe lines that intersect each other with the product of the slopes of lines -1 and different y-intercepts
\nIntersecting lines:
\nThe lines that are neither parallel nor perpendicular are called “Intersecting lines”
\nNote:
\nAll perpendicular lines should be intersecting lines but all intersecting lines should not be perpendicular lines<\/p>\n

Focus on math practices
\nLook for Relationships Is it possible for any of the pairs of lines drawn to have exactly two points in common? Explain.
\nAnswer:
\nWe know that,
\nAny pair of lines should intersect at only 1 common point
\nHence, from the above,
\nWe can conclude that It is not possible for any of the pairs of lines drawn to have exactly two points in common<\/p>\n

Essential Question<\/strong>
\nHow are slopes and y-intercepts related to the number of solutions of a system of linear equations?
\nAnswer:
\nWhen slopes and y-intercepts are different, the two lines intersect at one point,
\nThere is only 1 solution.
\nWhen\u00a0slopes\u00a0are the same, the lines may be either parallel (different\u00a0y-intercepts)
\nThere are no solutions
\nIf the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
\nSo, there are infinitely many solutions<\/p>\n

Try It!<\/strong><\/p>\n

How many solutions does this system of equations have? Explain.
\ny = x + 1
\ny = 2x + 2
\nThe system of equations has ________ solution. The equations have _______ slopes, so the lines intersect at ________ point.
\nAnswer:
\nThe given system of equations are:
\ny = x + 1 —–(1)
\ny = 2x + 2 —-(2)
\nCompare the given system of equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = 1, b = 1
\nFor the second equation,
\nm = 2 and b = 2
\nNow,
\nWhen we observe the slopes and y-intercepts, they are different
\nWe know that,
\nWhen slopes and y-intercepts are different, the two lines intersect at one point,
\nThere is only 1 solution.
\nHence, from the above,
\nWe can conclude that
\nThe given system of equations have only 1 solution
\nThe equation has 2 slopes
\nThe lines intersect at (-1, 0)<\/p>\n

Convince Me!
\nThe equations of a system have the same slopes. What can you determine about the solution of the system of equations?
\nAnswer:
\nIt is given that the equations of a system have the same slopes
\nHence,
\nWhen slopes are the same and the y-intercepts are different, the lines may be either parallel
\nSo,
\nThere are no solutions
\nIf the lines have the same slopes and the same y-intercepts, they intersect at all points,
\nSo,
\nThere are infinitely many solutions<\/p>\n

\"Analyze<\/p>\n

Try It!<\/strong><\/p>\n

How many solutions does each system of equations have? Explain.
\na. y = -3x + 5
\ny = -3x – 5
\nAnswer:
\nThe given system of equations are:
\ny = -3x + 5 ——(1)
\ny = -3x – 5 ——-(2)
\nCompare the given system of equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = -3, b = 5
\nFor the second equation,
\nm = -3, b = -5
\nNow,
\nFrom the above,
\nWe can observe that there are the same slopes but different y-intercepts
\nWe know that,
\nWhen slopes are the same and the y-intercepts are different, the lines may be either parallel and there are no solutions
\nHence, from the above,
\nWe can conclude that there are no solutions for the given system of equations<\/p>\n

b. y = 3x + 4
\n5y \u2013 15x \u2013 20 = 0
\nAnswer:
\nThe given system of equations are:
\ny = 3x + 4
\n5y – 15x – 20 = 0
\nNow,
\n5y = 15x + 20
\nDivide by 5 into both sides
\nSo,
\ny = 3x + 4
\nSo,
\nThe given system of equations are:
\ny = 3x + 4 ——-(1)
\ny = 3x + 4 ——–(2)
\nNow,
\nCompare the given system of equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = 3, b = 4
\nFor the second equation,
\nm = 3, b = 4
\nNow,
\nFrom the above,
\nWe can observe that there are the same slopes and the same y-intercepts
\nWe know that,
\nWhen slopes and the y-intercepts are the same,
\nThe lines have infinitely many solutions
\nHence, from the above,
\nWe can conclude that there are infinitely many solutions for the given system of equations<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

You can inspect the slopes and y-intercepts of the equations in a system of linear equations in order to determine the number of solutions of the system.
\nOne Solution
\ny = 2x + 4
\ny = 3x – 1
\nThe slopes are different. The lines intersect at 1 point.<\/p>\n

No Solution
\ny = 3x + 4
\ny = 3x + 5
\nThe slopes are the same, and the y-intercepts are different. The lines are parallel.<\/p>\n

Infinitely Many Solutions
\ny = 3x + 4
\ny = 4 + 3x
\nThe slopes are the same, and the y-intercepts are the same. The lines are the same.<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How are slopes and y-intercepts related to the number of solutions of a system of linear equations?
\nAnswer:
\nWhen slopes and y-intercepts are different, the two lines intersect at one point,
\nThere is only 1 solution.
\nWhen\u00a0slopes\u00a0are the same, the lines may be either parallel (different\u00a0y-intercepts)
\nThere are no solutions
\nIf the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
\nSo, there are infinitely many solutions<\/p>\n

Question 2.
\nConstruct Arguments<\/strong> Macy says that any time the equations in a system have the same y-intercept, the system has infinitely many solutions. Is Macy correct? Explain.
\nAnswer:
\nIt is given that
\nMacy says that any time the equations in a system have the same y-intercept, the system has infinitely many solutions.
\nNow,
\nConsider two equations with the same y-intercept and different slopes.
\ny = 2x + 3
\ny = 5x + 3
\nCompare the above system of equation with
\ny = mx + b
\nHence,
\nWe can say that
\nThis system has only one solution.
\nNow,
\nConsider another system of equations with the same y-intercepts and the same slopes
\ny = x + 7
\n<\/span>y = x + 7
\nCompare the above system of equation with
\ny = mx + b
\nHence,
\nWe can say that
\nThis system has infinitely many solutions.
\nHence, from the above,
\nWe can conclude that Macy is not correct<\/p>\n

Question 3.
\n<\/span>Use Structure How can you determine the number of solutions of a system of linear equations by inspecting its equations?
\nAnswer:
\nBy inspecting the equations, the number of solutions can be determined as mentioned below:
\nWhen slopes and y-intercepts are different, the two lines intersect at one point,
\nThere is only 1 solution.
\nWhen\u00a0slopes\u00a0are the same, the lines may be either parallel (different\u00a0y-intercepts)
\nThere are no solutions
\nIf the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
\nSo, there are infinitely many solutions<\/p>\n

\"Analyze<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nKyle has x 3-ounce blue marbles and a 5-ounce green marble. Lara has x 5-ounce green marbles and a 3-ounce blue marble. Is it possible for Kyle and Lara to have the same number of green marbles and the same total bag weight, y? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nKyle has x 3-ounce blue marbles and a 5-ounce green marble. Lara has x 5-ounce green marbles and a 3-ounce blue marble.
\nNow,
\nThe total number of marbles = The total number of green marbles + The total number of blue marbles
\nSo,
\nFor Kyle,
\nThe total number of marbles = x+ 1
\nFor Lara,
\nThe total number of marbles = x + 1
\nNow,
\nFrom the total number of marbles,
\nWe can observe that Kyle and Lara have the same number of marbles
\nSince Kyla and Lara have the same number of marbles,
\nThe total weight of the marbles will also be the same
\nHence, from the above,
\nWe can conclude that
\nKyle and Lara have the same number of green marbles and the same total bag weight, y<\/p>\n

Question 5.
\nHow many solutions does this system of linear equations have? Explain.
\n\\(\\frac{1}{2}\\)x = y
\ny = \\(\\frac{1}{2}\\)x + 3
\nAnswer:
\nThe given system of equations are:
\n\\(\\frac{1}{2}\\)x = y ——(1)
\ny = \\(\\frac{1}{2}\\)x + 3 —–(2)
\nNow,
\nComapre the given system of equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = \\(\\frac{1}{2}\\), b = 0
\nFor the second equation,
\nm = \\(\\frac{1}{2}\\), b = 3
\nWe know that,
\nWhen\u00a0slopes\u00a0are the same, the lines may be either parallel (different\u00a0y-intercepts)
\nThere are no solutions
\nHence, from the above,
\nWe can conclude that the given system of equations have no solutions<\/p>\n

Question 6.
\nHow many solutions does this system of linear equations have? Explain.
\n3y + 6x = 12
\n8x + 4y = 16
\nAnswer:
\nThe given system of equations are:
\n3y + 6x = 12
\n8x + 4y = 16
\nSo,
\n3y = -6x + 12
\nDivide by 3 into both sides
\nSo,
\ny = -2x + 4
\nSo,
\n4y = -8x + 16
\nDivide by 4 into both sides
\nSo,
\ny = -2x + 4
\nSo,
\nThe required system of equations are:
\ny = -2x + 4 —–(1)
\ny = -2x + 4 —–(2)
\nWe know that,
\nIf the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
\nSo, there are infinitely many solutions
\nHence, from the above,
\nWe can conclude that the given system of equations have infinitely many solutions<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nLeveled Practice Two rovers are exploring a planet. The system of equations below shows each rover’s elevation, y, at time x. What conclusion can you reach about the system of equations?
\n\"Envision
\nRover A: y = 1.9x – 8
\nRover B: 7y = 13.3x – 56
\nThe slope for the Rover A equation is _________ the slope for the Rover B equation.
\nThe y-intercepts of the equations are ___________.
\nThe system of equations has __________ solution(s).
\nAnswer:
\nIt is given that
\nTwo rovers are exploring a planet. The system of equations below shows each rover’s elevation, y, at time x.
\nNow,
\nThe given system of equation are:
\nRover A: y = 1.9x – 8 ——(1)
\nRover B: 7y = 13.3x – 56
\nNow,
\nDivide the equation of Rover B with 7
\nSo,
\nRover B: y = 1.9x – 8 ——(2)
\nNow,
\nWhen we compare the given system of equations with
\ny = mx + b
\nFor Rover A,
\nm = 1.9, b = -8
\nFor Rover B,
\nm = 1.9, b = -8
\nWe know that,
\nIf the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
\nSo, there are infinitely many solutions
\nHence, from the above,
\nWe can conclude that the given system of equations ahve infinitely many solutions<\/p>\n

Question 8.
\nHow many solutions does this system have?
\ny = x – 3
\n4x \u2013 10y = 6
\nAnswer:
\nThe given system of equations are:
\ny = x – 3 ——(1)
\n4x – 10y = 6
\nSo,
\n10y = 4x – 6
\nDivide by 10 into both sides
\nSo,
\ny = 0.4x – 0.6 ——(2)
\nNow,
\nCompare the given system of equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = 1, b = -3
\nFor the second equation,
\nm = 0.4, b = -0.6
\nWe know that,
\nWhen slopes and y-intercepts are different, the two lines intersect at one point,
\nThere is only 1 solution.
\nHence, from the above,
\nWe can conclude that the given system of equations have only 1 solution<\/p>\n

Question 9.
\nHow many solutions does this system have?
\nx + 3y = 0
\n12y = -4x
\nAnswer:
\nThe given system of equations are:
\nx + 3y = 0
\n12y = -4x
\nNow,
\n3y = -x + 0
\ny = –\\(\\frac{1}{3}\\)x + 0 ——(1)
\n12y = -4x
\ny = –\\(\\frac{4}{12}\\)x
\ny = –\\(\\frac{1}{3}\\)x ——-(2)
\nNow,
\nCompare the given system of equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = –\\(\\frac{1}{3}\\), b = 0
\nFor the second equation,
\nm = –\\(\\frac{1}{3}\\), b = 0
\nWe know that,
\nIf the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
\nSo, there are infinitely many solutions
\nHence, from the above,
\nWe can conclude that the given system of equations ahve infinitely many solutions<\/p>\n

Question 10.
\nWhat can you determine about the solution(s) of this system?
\n-64x + 96y = 176
\n56x – 84y = -147
\nAnswer:
\nThe given system of equations are:
\n-64x + 96y = 176
\n56x – 84y = -147
\nSo,
\n96y = 64x + 176
\n84y = 56x + 147
\nNow,
\nDivide by 96 into both sides
\ny = \\(\\frac{2}{3}\\)x + \\(\\frac{11}{6}\\) ——(1)
\nNow,
\nDivide by 84 into both sides
\ny = \\(\\frac{2}{3}\\)x + \\(\\frac{7}{4}\\) ——-(2)
\nNow,
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = \\(\\frac{2}{3}\\), b = \\(\\frac{11}{6}\\)
\nFor the second equation,
\nm = \\(\\frac{2}{3}\\), b = \\(\\frac{7}{4}\\)
\nWe know that,
\nWhen\u00a0slopes\u00a0are the same, the lines may be either parallel (different\u00a0y-intercepts)
\nThere are no solutions
\nHence, from the above,
\nWe can conclude that the given system of equations have no solutions<\/p>\n

Question 11.
\nDetermine whether this system of equations has one solution, no solution, or infinitely many solutions.
\ny = 8x + 2
\ny = -8x + 2
\nAnswer:
\nThe given system of equations are:
\ny = 8x + 2 —–(1)
\ny = -8x + 2 —–(2)
\nNow,
\nCompare the given system of equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = 8, b = 2
\nFor the second equation,
\nm = -8, b = 2
\nWe know that,
\nWhen slopes and y-intercepts are different or y-intercepts are the same, the two lines intersect at one point,
\nThere is only 1 solution.
\nHence, from the above,
\nWe can conclude that the given system of equations have only 1 solution<\/p>\n

Question 12.
\nConstruct Arguments<\/strong> Maia says that the two lines in this system of linear equations are parallel. Is she correct? Explain.
\n2x + y = 14
\n2y + 4x = 14
\nAnswer:
\nIt is given that
\nMaia said the below system of linear equations to be paralle
\nNow,
\nThe given system of equations are:
\n2x + y = 14
\n2y + 4x = 14
\nNow,
\ny = -2x + 14 —-(1)
\n2y = -4x + 14
\nDivide by 2 into both sides
\nSo,
\ny = -2x + 7 —–(2)
\nNow,
\nCompare the given system of equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = -2, b = 14
\nFor the second equation,
\nm = -2, b = 7
\nWe know that,
\nFor the lines to be parallel, the slopes have to be the same and the y-intercepts be different
\nHence, from the above,
\nWe can conclude that Maia is correct<\/p>\n

Question 13.
\nReasoning<\/strong> Describe a situation that can be represented by using this system of equations. Inspect the system to determine the number of solutions and interpret the solution within the context of your situation.
\ny = 2x + 10
\ny = x + 15
\nAnswer:
\nThe given system of equations are:
\ny = 2x + 10 — (1)
\ny = x + 15 —–(2)
\nNow,
\nCompare the above system of equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = 2, b = 10
\nFor the second equation,
\nm = 1, b = 15
\nWe know that,
\nWhen the slopes and y-intercepts are different for a system of linear equations,
\nThere is only 1 solution
\nHence, from the above,
\nWe can conclude that the given system of equations has only 1 solution<\/p>\n

Question 14.
\nLook for Relationships Does this system have one solution, no solutions, or infinitely many solutions? Write another system of equations with the same number of solutions that uses the first equation only.
\n12x + 51y = 156
\n-8x \u2013 34y = -104
\nAnswer:
\nThe given system of equations are:
\n12x + 51y = 156
\n-8x – 34y = -104
\nSo,
\n51y = -12x + 156
\nDivide by 51 into both sides
\nSo,
\ny = –\\(\\frac{4}{17}\\)x + \\(\\frac{52}{17}\\) —-(1)
\nSo,
\n8x + 34y = 104
\n34y = -8x + 104
\nDivide by 34 into both sides
\nSo,
\ny = –\\(\\frac{4}{17}\\)x + \\(\\frac{52}{17}\\) ——-(2)
\nNow,
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = –\\(\\frac{4}{17}\\), b = \\(\\frac{52}{17}\\)
\nFor the second equation,
\nm = –\\(\\frac{4}{17}\\), b = \\(\\frac{52}{17}\\)
\nWe know that,
\nWhen we have the same slopes and the same y-intercepts, the solutions for a system of linear equations are infinite
\nNow,
\nAnother system of equations with the first equation as one of the equations is:
\n12x + 51y = 156
\n48x + 204y = 624
\nHence, from the above,
\nWe can conclude that there are infinitely many solutions for the given system of equations<\/p>\n

Question 15.
\nThe equations represent the heights, y, of the flowers, in inches, after x days. What does the y-intercept of each equation represent? Will the flowers ever be the same height? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe equations represent the heights, y, of the flowers, in inches, after x days.
\nNow,
\nThe given system of equations are:
\ny = 0.7x + 2 —–(1)
\ny = 0.4x + 2 —–(2)
\nNow,
\nCompare the above system of equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = 0.7, b = 2
\nFor the second equation,
\nm = 0.4, b = 2
\nNow,
\nWe know that
\nb represents the y-intercept or the initial value
\nNow,
\nWhen we compare the slopes of the equations,
\nWe know that,
\nThe slopes are different
\nHence, from the above,
\nWe can conclude that
\nThe y-intercept of each equation represents the initial height of the flowers
\nThe heights of the flowers are not the same<\/p>\n

Question 16.
\nDoes this system have one solution, no solution, or an infinite number of solutions?
\n4x + 3y = 8
\n8x + y = 2
\nAnswer:
\nThe given system of equations are:
\n4x + 3y = 8
\n8x + y = 2
\nNow,
\n3y = -4x + 8
\nDivide by 3 into both sides
\nSo,
\ny = –\\(\\frac{4}{3}\\)x + \\(\\frac{8}{3}\\) —-(1)
\nSo,
\ny = -8x + 2 —–(2)
\nNow,
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFor the equation 1,
\nm = –\\(\\frac{4}{3}\\), b = \\(\\frac{8}{3}\\)
\nFor the equation 2,
\nm = -8, b = 2
\nWe know that,
\nWhen the slopes and y-intercepts of a system of equations are different, the system has only a solution
\nHence, from the above,
\nWe can conclude that the given system of equations has only 1 solution<\/p>\n

Question 17.
\nHigher-Order Thinking<\/strong> Under what circumstances does the system of equations Qx + Ry = S and y = Tx + S have infinitely many solutions?
\nAnswer:
\nThe given system of equations are:
\nQx + Ry = S
\ny = Tx + S
\nSo,
\nRy = -Qx + S
\nDivide by R into both sides
\nSo,
\ny = –\\(\\frac{Q}{R}\\)x + \\(\\frac{S}{R}\\) —–(1)
\ny = Tx + S —–(2)
\nNow,
\nFor the given system of equations to have infinitely many solutions,
\nThe slopes must be the same
\nThe y-intercepts must be the same
\nSo,
\n–\\(\\frac{Q}{R}\\) = T
\n\\(\\frac{S}{R}\\) = S
\nHence, from the above,
\nWe can conclude that the given system of equations will have infinitely many solutions when
\n–\\(\\frac{Q}{R}\\) = T
\n\\(\\frac{S}{R}\\) = S<\/p>\n

Assessment Practice<\/strong>
\nQuestion 18.
\nBy inspecting the equations, what can you determine about the solution(s) of this system?
\n12y = 9x + 33
\n20y = 15x + 55
\nAnswer:
\nThe given system of equations are:
\n12y = 9x + 33 —-(1)
\n20y = 15x + 55 —-(2)
\nSo,
\nDivide eq (1) by 12 into both sides
\nSo,
\ny = \\(\\frac{3}{4}\\)x + \\(\\frac{11}{4}\\) —-(3)
\nSo,
\nDivide eq (2) by 20 into both sides
\nSo,
\ny = \\(\\frac{3}{4}\\)x + \\(\\frac{11}{4}\\) —–(4)
\nNow,
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFor the 3rd equation,
\nm = \\(\\frac{3}{4}\\), b = \\(\\frac{11}{4}\\)
\nFor the 4th equation,
\nm = \\(\\frac{3}{4}\\), b = \\(\\frac{11}{4}\\)
\nWe know that,
\nWhen the slopes and the y-intercepts of a system of equations are different, the system has only 1 solution
\nHence, from the above,
\nWe can conclude that the given system of equations has infinitely many solutions<\/p>\n

Question 19.
\nChoose the statement that correctly describes how many solutions there are for this system of equations.
\ny = \\(\\frac{2}{3}\\)x + 3
\ny = \\(\\frac{5}{4}\\)x + 3
\nA. Infinitely many solutions because the slopes are equal and the y-intercepts are equal
\nB. Exactly one solution because the slopes are equal but the y-intercepts are NOT equal
\nC. No solution because the slopes are equal and the y-intercepts are NOT equal
\nD. Exactly one solution because the slopes are NOT equal
\nAnswer:
\nThe given system of equations are:
\ny = \\(\\frac{2}{3}\\)x + 3 —-(1)
\ny = \\(\\frac{5}{4}\\)x + 3 —–(2)
\nNow,
\nCompare the abovee quations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = \\(\\frac{2}{3}\\), b = 3
\nFpr the second equation,
\nm = \\(\\frac{5}{4}\\), b = 3
\nWe know that
\nWhen the slopes are different and the y-intercepts are the same or different, the system of equations has only 1 solution
\nHence, from the above,
\nWe can conclude that option D matches with the given system of equations<\/p>\n

Lesson 5.2 Solve Systems by Graphing<\/h3>\n

Explore It!<\/strong>
\nBeth and Dante pass by the library as they walk home using separate straight paths.
\n\"Envision
\nI can… find the solution to a system of equations using graphs.<\/p>\n

A. Model with Math The point on the graph represents the location of the library. Draw and label lines on the graph to show each possible path to the library.
\n\"Envision
\nAnswer:
\nIt is given that
\nBeth and Dante pass by the library as they walk home using separate straight paths.
\nNow,
\nIn the graph,
\nthe location of the graph is given
\nHence,
\nThe representation of one of the paths of Beth and Dante that are passing through the library in a straight line is:
\n\"\"<\/p>\n

B. Write a system of equations that represents the paths taken by Beth and Dante.
\nAnswer:
\nFrom part (a),
\nThe representation of one of the paths of Beth and Dante that are passing through the library in a straight line is:
\n\"\"
\nSo,
\nFrom the graph,
\nThe points that are passing through the path of Beth is: (-4, 0), (0, 2)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope (m) = \\(\\frac{2 – 0}{0 + 4}\\)
\n= \\(\\frac{2}{4}\\)
\n= \\(\\frac{1}{2}\\)
\nWe know that,
\nThe form of the linear equation in the slope-intercept form is:
\ny = mx + b
\nSo,
\ny = \\(\\frac{1}{2}\\)x + b
\n2y = x + 2b
\nSubstitute (-4, 0) (or) (0, 2) in the above equation
\nSo,
\n2b = 4
\nb = 2
\nSo,
\nThe required equation is:
\n2y = x + 4
\nNow,
\nThe points that are passing through the path of Dante is: (-1, -2), (0, 0)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nSlope (m) = \\(\\frac{0 + 2}{0 + 1}\\)
\n= \\(\\frac{2}{1}\\)
\n= 2
\nWe know that,
\nThe form of the linear equation in the slope-intercept form is:
\ny = mx + b
\nSo,
\ny = 2x + b
\nSubstitute (-1, -2) (or) (0, 0) in the above equation
\nSo,
\nb = 0
\nSo,
\nThe required equation is:
\ny = 2x
\nHence, from the above,
\nWe can conclude that
\nThe required system of linear equations are:
\n2y = x + 4
\ny = 2x<\/p>\n

Focus on math practices
\nReasoning What does the point of intersection of the lines represent in the situation?
\nAnswer:
\nPoint of intersection means the point\u00a0at which two\u00a0lines intersect. These two\u00a0lines are represented\u00a0by the equation
\na1<\/sub>x + b1<\/sub>y + c1<\/sub>= 0 and
\na2<\/sub>x + b2<\/sub>y + c2<\/sub>\u00a0= 0, respectively.
\nBy solving the two equations, we\u00a0can\u00a0find the solution for the\u00a0point of intersection\u00a0of two\u00a0lines.<\/p>\n

Essential Question<\/strong>
\nHow does the graph of a system of linear equations represent its solution?
\nAnswer:
\nEach shows two lines that make up a system\u00a0of\u00a0equations. If the\u00a0graphs\u00a0of the\u00a0equations\u00a0intersect, then there is one\u00a0solution\u00a0that is true for both\u00a0equations. If the\u00a0graphs\u00a0of the\u00a0equations do\u00a0not intersect (for example, if they are parallel), then there are no\u00a0solutions\u00a0that are true for both\u00a0equations.<\/p>\n

Try It!<\/strong><\/p>\n

Solve the system by graphing.
\n\"Envision
\ny = 3x + 5
\ny = 2x + 4
\nThe solution is the point of intersection (______, _______)
\nAnswer:
\nThe given system of equations are:
\ny = 3x + 5
\ny = 2x + 4
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution or the intersection point of the given system of equations is: (-1, 2)<\/p>\n

Convince Me!
\nHow does the point of intersection of the graphs represent the solution of a system of linear equations?
\nAnswer:
\nWhen you graph an equation, each point (x,y) on the line satisfies the equation. Therefore, when 2 lines intersect, the coordinates of the intersection point satisfy both equations, i.e. the\u00a0intersection point represents the solution\u00a0of the set<\/p>\n

Try It!<\/strong><\/p>\n

Solve each system by graphing. Describe the solutions.
\n\"Envision
\na. 5x + y = -3
\n10x + 2y = -6
\nAnswer:
\nThe given system of equations are:
\n5x + y = -3
\n10x + 2y = -6
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the given system of equations has infinitely many solutions since both equations are on the same line<\/p>\n

b. x + y = 7
\n2x + 6y = 12
\nAnswer:
\nThe given system of equations are:
\nx + y = 7
\n2x + 6y = 12
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the given system of equations has only one solution i.e., (7.5, -0.5)<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

The solution of a system of linear equations is the point of intersection of the lines defined by the equations.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How does the graph of a system of linear equations represent its solution?
\nAnswer:
\nEach shows two lines that make up a system\u00a0of\u00a0equations. If the\u00a0graphs\u00a0of the\u00a0equations\u00a0intersect, then there is one\u00a0solution\u00a0that is true for both\u00a0equations. If the\u00a0graphs\u00a0of the\u00a0equations do\u00a0not intersect (for example, if they are parallel), then there are no\u00a0solutions\u00a0that are true for both\u00a0equations.<\/p>\n

Question 2.
\nReasoning<\/strong> If a system has no solution, what do you know about the lines being graphed?
\nAnswer:
\nIf a system has no solution, it is said to be inconsistent. The\u00a0graphs of\u00a0the\u00a0lines do not\u00a0intersect, so the\u00a0graphs are\u00a0parallel and there\u00a0is no solution.<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> in a system of linear equations, the lines described by each equation have the same slopes. What are the possible solutions to the system? Explain.
\nAnswer:
\nIf the given system of equations has the same slope, then
\nFor the same slope and the same y-intercept – There ate infinitely many solutions
\nFor the same slope and different outputs – There are no solutions<\/p>\n

Do You Know How?<\/strong>
\nIn 4-6, graph each system of equations and find the solution.
\nQuestion 4.
\ny = -3x – 5
\ny = 9x + 7
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\ny = -3x – 5
\ny = 9x + 7
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (-1, -2)<\/p>\n

Question 5.
\ny = 2x – 5
\n6x + 3y = -15
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\ny = 2x – 5
\n6x + 3y = -15
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (0, -5)<\/p>\n

Question 6.
\ny = -4x + 3
\n8x + 2y = 8
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\ny = -4x + 3
\n8x + 2y = 8
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that there are no solutions for the given system of equations<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

In 7 and 8, graph each system of equations to determine the solution.
\nQuestion 7.
\nx + 4y = 8
\n3x + 4y = 0
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\nx + 4y = 8
\n3x + 4y = 0
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (-4, 3)<\/p>\n

Question 8.
\n2x \u2013 3y = 6
\n4x – y = 12
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\n2x – 3y = 6
\n4x – y = 12
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (3, 0)<\/p>\n

Question 9.
\nThe total cost, c, of renting a canoe for n hours can be represented by a system of equations.
\n\"Envision
\na. Write the system of equations that could be used to find the total cost, c, of renting a canoe for n hours.
\nAnswer:
\nIt is given that
\nThe total cost, c, of renting a canoe for n hours can be represented by a system of equations.
\nNow,
\nThe total cost for renting a canoe (y) = The cost of renting a canoe per hour \u00d7 The number of hours (n) + Deposit
\nSo,
\nFor River Y,
\ny = 33n
\nFor River Z,
\ny = 5n + 13
\nHence, from the above,
\nWe can conclude that the system of equations for the total cost of a canoe is:
\ny = 33n
\ny = 5n + 13<\/p>\n

b. Graph the system of equations.
\n\"Envision
\nAnswer:
\nFrom part (a),
\nThe system of equations is:
\ny = 33n
\ny = 5n + 13
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution of the given system of equations is: (0.5, 16)<\/p>\n

c. When would the total cost for renting a canoe be the same on both rivers? Explain.
\nAnswer:
\nFrom the graph that is in part (b),
\nWe can observe that the y-axis passes through 12
\nHence,from the above,
\nWe can conclude that the total cost for renting a canoe can be the same on both rivers after 12 hours<\/p>\n

Question 10.
\nGraph the system of equations and determine the solution.
\nx + 2y = 4
\n4x + 8y = 64
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\nx + 2y = 4
\n4x + 8y = 64
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that there are no solutions for the given system of equations since they are parallel<\/p>\n

Question 11.
\nGraph the system of equations, then estimate the solution.
\ny = 1.5x + 1
\ny = -1.5x + 5.5
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\ny = 1.5x + 1
\ny = -1.5x + 5.5
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the estimated solution for the given system of equations is: (1, 3)<\/p>\n

In 12 and 13, graph and determine the solution of the system of equations.
\nQuestion 12.
\n-3y = -9x + 3
\n-6y = -18x \u2013 12
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\n-3y = -9x + 3
\n-6y = -18x – 12
\nSo,
\nThe representation of the given syetem of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that there are no solutions for the given system of equations since they are parallel lines<\/p>\n

Question 13.
\nx + 5y = 0
\n25y = -5x
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\nx + 5y = 0
\n25y = -5x
\nSo,
\nThe representation of the given system of equations i the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that there are infinitely many solutions for the given system of equations since both equations are in the same line<\/p>\n

Question 14.
\nHigher Order Thinking<\/strong> The total cost, c, of making n copies can be represented by a system of equations.
\n\"Envision
\na. Estimate how many copies you need to make for the total cost to be the same at both stores.
\nAnswer:
\nIt is given that
\nThe total cost, c, of making n copies can be represented by a system of equations.
\nWe know that,
\n1 dollar = 0.1 cent
\nSo,
\nThe total cost of making n copies (y) = The number of copies per hour \u00d7 The number of copies + the cost of machine use
\nSo,
\nFor Store W,
\ny = 5n —— (1)
\nFor Store Z,
\ny = 0.20n + 2 —- (2)
\nNow,
\nFrom the given graph,
\nWe can observe that the given system of equations pass through (6, 5)
\nHence, from the above,
\nWe can conclude that we have to make 6 copies for the total cost to be the same at both stores<\/p>\n

b. If you have to make a small number of copies, which store should you go to? Explain.
\nAnswer:
\nTo make a small number of copies,
\nWe will see which store gives us the less cost for printing the number of copies
\nSo,
\nFrom the given information,
\nWe can observe that Store Z gives us the less cost for printing the more number of copies
\nHence, from the above,
\nWe can conclude that we will goto Store Z if we have to make small number of copies<\/p>\n

Assessment Practice<\/strong>
\nQuestion 15.
\nConsider the following system of equations.
\ny =-3x + 6
\ny = 3x – 12
\nWhich statement is true about the system?
\nA. The graph of the system is a pair of lines that do not intersect.
\nB. The graph of the system is a pair of lines that intersect at exactly one point.
\nC. The graph of the system is a pair of lines that intersect at every point.
\nD. The system has infinitely many solutions.
\nAnswer:
\nThe given system of equations are:
\ny = -3x + 6
\ny = 3x – 12
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that option B matches with the description of the above graph<\/p>\n

Question 16.
\nWhat is the solution of the system of equations?
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nSo,
\nFrom the given graph,
\nWe can observe that
\nThe intersection point is: (2.5, -3)
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations are: (2.5, 3)<\/p>\n

Topic 5 MID-TOPIC CHECKPOINT<\/h3>\n

Question 1.
\nVocabulary How can you determine the number of solutions of a system by looking at the equations? Lesson 5-1
\nAnswer:
\nA\u00a0system\u00a0of two\u00a0equations\u00a0can be classified as follows: If the slopes are the same but the y-intercepts are different, the\u00a0system has\u00a0no\u00a0solution. If the slopes are different, the\u00a0system has\u00a0one\u00a0solution. If the slopes are the same and the y-intercepts are the same, the\u00a0system has\u00a0infinitely\u00a0many solutions.<\/p>\n

Question 2.
\nHow many solutions does the system of equations have? Explain. Lesson 5-1
\n2x – 9y = -5
\n4x – y = 2
\nAnswer:
\nThe given system of equations are:
\n2x – 9y = -5
\n4x – y = 2
\nSo,
\n9y = 2x + 5
\ny = \\(\\frac{2}{9}\\)x + \\(\\frac{5}{9}\\) —— (1)
\nSo,
\ny = 4x – 2 —— (2)
\nNow,
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = \\(\\frac{2}{9}\\), b = \\(\\frac{5}{9}\\)
\nFor the second equation,
\nm = 4, b = -2
\nWe know that,
\nWhen the slopes and y-intercepts of a given system of equations are different, the system of equations has only one solution
\nHence, from the above,
\nWe can conclude that the given system of equations has only one solution<\/p>\n

Question 3.
\nGraph the system of equations and find the solution. Lesson 5-2
\ny = 2x \u2013 1
\ny = \\(\\frac{1}{2}\\)x + 2
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\ny = 2x – 1
\ny = \\(\\frac{1}{2}\\)x + 2
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution of the given system of equations is: (2, 3)<\/p>\n

Question 4.
\nOne equation in a system is y =-3x + 7. Which equation gives the system no solution? Lesson 5-1
\nA. y = -3x + 7
\nB. y = 3x + 5
\nC. y = -3x + 5
\nD. y= \\(\\frac{1}{3}\\)x – 7
\nAnswer:
\nIt is given that
\nOne of the equations in a system of equations is:
\ny = -3x + 7
\nNow,
\nFor a system of equations to have no solution,
\nThe slopes must be equal but the y-intercepts must be different
\nSo,
\nWith the above description, we have 2 options matched but option A has the same slope and y-intercept as the first equation
\nHence, from the above,
\nWe can conclude that option C must be the other equation in the given system of equations<\/p>\n

Question 5.
\nFinn bought 12 movie tickets. Student tickets cost $4, and adult tickets cost $8. Finn spent a total of $60. Write and graph a system of equations to find the number of student and adult tickets Finn bought. Lesson 5-2
\n\"Envision
\nAnswer:
\nIt is given that
\nFinn bought 12 movie tickets. Student tickets cost $4, and adult tickets cost $8. Finn spent a total of $60.
\nNow,
\nLet x be the number of student tickets
\nLet y be the number of adult tickets
\nSo,
\nx + y = 12 —– (1) [The total number of tickets]
\nSo,
\n4x + 8y = 60 —– (2) [The total cost of the tickets]
\nSo,
\nThe representation of the above equations in the coordinate plane is:
\n\"\"
\nSo,
\nFrom the above graph,
\nThe intersection point of the graph is: (9, 3)
\nHence, from the above,
\nWe can conclude that
\nThe number of student tickets is: 9
\nThe number of adult tickets is: 3<\/p>\n

Question 6.
\nWhat value of m gives the system infinitely many solutions? Lesson 5-1
\n-x + 4y = 32
\ny = mx + 8
\nAnswer:
\nThe given system of equations are:
\n-x + 4y = 32
\ny = mx + 8
\nSo,
\n4y = x + 32
\ny = \\(\\frac{1}{4}\\)x + 8 —– (1)
\ny = mx + 8 ——- (2)
\nNow,
\nFor the given system of equations to have infinitely many solutions,
\nThe slopes and the y-intercepts of the 2 equations must be equal
\nSo,
\nm = \\(\\frac{1}{4}\\)
\nHence, from the above,
\nWe can conclude that the value of m is: \\(\\frac{1}{4}\\)<\/p>\n

Topic 5 MID-TOPIC PERFORMANCE TASK<\/h3>\n

Perpendicular lines intersect to form right angles. The system of equations below shows perpendicular lines.
\n\"Envision<\/p>\n

PART A
\nHow many solutions does the system have? Explain.
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that there is only 1 intersection point
\nHence, from the above,
\nWe can conclude that the given system of equations has only 1 solution<\/p>\n

PART B
\nIdentify the slope and y-intercept of each line. What do you notice about the slopes of the lines?
\nAnswer:
\nFrom the given graph,
\nThe system of equations is:
\ny = \\(\\frac{3}{4}\\)x + 1 —— (1)
\ny = –\\(\\frac{4}{3}\\)x – 2 —— (2)
\nNow,
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = \\(\\frac{3}{4}\\), b = 1
\nFor the second equation,
\nm = –\\(\\frac{4}{3}\\), b = -2
\nNow,
\nFrom the slopes of the 2 equations,
\nWe can observe that the product of the 2 slopes are equal to -1<\/p>\n

PART C
\nWhat value of m makes the system show perpendicular lines? Explain.
\ny = \\(\\frac{1}{2}\\)x + 8
\ny = mx – 6
\nAnswer:
\nThe given system of equations are:
\ny = \\(\\frac{1}{2}\\)x + 8 —- (1)
\ny = mx – 6 —— (2)
\nNow,
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = \\(\\frac{1}{2}\\), b = 8
\nFor the second equation,
\nm = m, b = -6
\nWe know that,
\nFor the system of equations to be perpendicular,
\nThe product of the slopes must be equal to -1
\nSo,
\nm1<\/sub> . m2<\/sub> = -1
\nSo,
\n\\(\\frac{1}{2}\\)m = -1
\nm = -2
\nHence, from the above,
\nWe ca conclude that the value of m is: -1<\/p>\n

Lesson 5.3 Solve Systems by Substitution<\/h3>\n

Explain It!<\/strong>
\nJackson needs a taxi to take him to a destination that is a little over 4 miles away. He has a graph that shows the rates for two companies. Jackson says that because the slope of the line that represents the rates for On-Time Cabs is less than the slope of the line that represents Speedy Cab Co., the cab ride from On-Time Cabs will cost less.
\n\"Envision
\nI can… solve systems of equations using substitution.<\/p>\n

A. Do you agree with Jackson? Explain.
\nAnswer:
\nIt is given that
\nJackson needs a taxi to take him to a destination that is a little over 4 miles away. He has a graph that shows the rates for two companies. Jackson says that because the slope of the line that represents the rates for On-Time Cabs is less than the slope of the line that represents Speedy Cab Co., the cab ride from On-Time Cabs will cost less.
\nNow,
\nWe know that,
\nFor a linear graph,
\nQuantity 1 (The component of the x-axis) \u221d Quantity 2 (The component of the y-axis)
\nSo,
\nFrom the given information,
\nSlope \u221d Cost
\nHence, from the above information,
\nWe can agree with Jackson<\/p>\n

B. Which taxi service company should Jackson call? Explain your reasoning.
\nAnswer:
\nWe know that,
\nThe taxi service company must be chosen according to the cost
\nHence, from the above,
\nWe can conclude that Jackson should call for an On-Time cabs taxi service since the cost is less<\/p>\n

Focus on math practices
\nBe Precise Can you use the graph to determine the exact number of miles for which the cost of the taxi ride will be the same? Explain.
\nAnswer:
\nFrom the given graph,
\nWe can observe that the intersection point is the point where the cost of the taxi ride will be the same
\nSo,
\nCorresponding to that intersection point, the distance will be calculated for the same cost of the taxi rides
\nHence, from the above,
\nWe can conclude that you can use the graph to determine the exact number of miles for which the cost of the taxi ride will be the same<\/p>\n

Essential Question<\/strong> When is substitution a useful method for solving systems of equations?
\nAnswer:
\nThe\u00a0substitution method\u00a0is most\u00a0useful\u00a0for\u00a0systems\u00a0of 2\u00a0equations\u00a0in 2 unknowns. The main idea here is that we\u00a0solve\u00a0one of the\u00a0equations\u00a0for one of the unknowns, and then\u00a0substitute\u00a0the result into the other\u00a0equation.<\/p>\n

Try It!<\/strong><\/p>\n

Brandon took a 50-question exam worth a total of 160 points. There were x two-point questions and y five-point questions. How many of each type of question were on the exam?
\nx + y = 50
\n2x + 5y = 160
\ny = _____ – ______
\nSubstitute for y: 2x + 5(_____ – _____) = 160
\n2x + _____ – ______x = 160
\nx = ______ two-point questions
\nSubstitute for x: _____ + y = 50
\ny = _____ five-point questions
\nAnswer:
\nIt is given that
\nBrandon took a 50-question exam worth a total of 160 points. There were x two-point questions and y five-point questions.
\nNow,
\nThe given system of equations are:
\nx + y = 50 —– (1)
\n2x + 5y = 160 —–(2)
\nSo,
\nFrom eq (1),
\ny = 50 – x
\nNow,
\nSubstitute y in eq (2)
\nSo,
\n2x + 5 (50 – x) = 160
\n2x + 5 (50) – 5x = 160
\n2x + 250 – 5x = 160
\n-3x = 160 – 250
\n-3x = -90
\n3x = 90
\nx = \\(\\frac{90}{3}\\)
\nx = 30
\nSo,
\ny = 50 – x
\ny = 50 – 30
\ny = 20
\nHence from the above,
\nWe can conclude that
\nThe number of 2-point questions is: 30
\nThe number of 5-point questions is: 20<\/p>\n

Convince Me!
\nHow do you know which equation to choose to solve for one of the variables?
\nAnswer:
\nThe idea of substitution is that if\u00a0one variable\u00a0lets you express\u00a0one variable\u00a0in terms of the other, you can substitute that expression for the\u00a0variable\u00a0in the other\u00a0equation. That way the second\u00a0equation\u00a0only has\u00a0one variable, and you can\u00a0solve\u00a0that<\/p>\n

Try It!<\/strong><\/p>\n

Use substitution to solve each system of equations. Explain.
\na. y + \\(\\frac{1}{2}\\)x = 3
\n4y + 2x = -6
\nAnswer:
\nThe given system of equations are:
\ny + \\(\\frac{1}{2}\\)x = 3 —- (1)
\n4y + 2x = -6 ——- (2)
\nSo,
\nFrom eq (1),
\ny = 3 – \\(\\frac{1}{2}\\)x
\nNow,
\nSubstitute y in eq (2)
\nSo,
\n4 (3 – \\(\\frac{1}{2}\\)x) + 2x = -6
\n4 (3) – 2x + 2x = -6
\n12 = -6
\nHence, from the above,
\nWe can conclude that there is no solution for the given system of equations<\/p>\n

b. y = \\(\\frac{1}{4}\\)x – 2
\n8y \u2013 2x = -16
\nAnswer:
\nThe given system of equations are:
\ny = \\(\\frac{1}{4}\\)x – 2 —– (1)
\n8y – 2x = -16
\nNow,
\nSubstitute eq (1) in eq (2)
\nSo,
\n8 (\\(\\frac{1}{4}\\)x – 2) – 2x = -16
\n2x – 8 (2) – 2x = -16
\n-16 = -16
\n16 = 16
\nHence, from the above,
\nWe can conclude that there are infinitely many solutions for the given system of equations<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

Systems of linear equations can be solved algebraically. When one of the equations can be easily solved for one of the variables, you can use substitution to solve the system efficiently.
\nSTEP 1 Solve one of the equations for one of the variables. Then substitute the expression into the other equation and solve.
\nSTEP 2 Solve for the other variable using either equation.<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> when is substitution a useful method for solving systems of equations?
\nAnswer:
\nThe\u00a0substitution method\u00a0is most\u00a0useful\u00a0for\u00a0systems\u00a0of 2\u00a0equations\u00a0in 2 unknowns. The main idea here is that we\u00a0solve\u00a0one of the\u00a0equations\u00a0for one of the unknowns, and then\u00a0substitute\u00a0the result into the other\u00a0equation.<\/p>\n

Question 2.
\nGeneralize when using substitution to solve a system of equations, how can you tell when a system has no solution?
\nAnswer:
\nWhen a\u00a0system has no solution\u00a0or an infinite number of\u00a0solutions\u00a0and\u00a0we\u00a0attempt to find a single, unique\u00a0solution using\u00a0an algebraic method, such as\u00a0substitution, the variables\u00a0will\u00a0cancel out and\u00a0we will have\u00a0an\u00a0equation\u00a0consisting of only constants.\u00a0If\u00a0the\u00a0equation is\u00a0untrue then the\u00a0system has no solution.<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Kavi solved the system of equations as shown. What mistake did Kavi make? What is the correct solution?
\n3x + 4y = 33
\n2x + y = 17
\ny = 17 – 2x
\n2x + (17 \u2013 2x) = 17
\n2x + 17 – 2x = 17
\n2x – 2x + 17 = 17
\n17 = 17
\nInfinitely many solutions
\nAnswer:
\nThe given system of equations are:
\n3x + 4y = 33 —– (1)
\n2x + y = 17 —– (2)
\nFrom eq (2),
\ny = 17 – 2x
\nNow,
\nSubstitute y in eq (1)
\nSo,
\n3x + 4 (17 – 2x) = 33
\n3x + 4 (17) – 4 (2x) = 33
\n3x + 68 – 8x = 33
\n-5x = 33 – 68
\n-5x = -35
\n5x = 35
\nx = \\(\\frac{35}{5}\\)
\nx = 7
\nHence, from the above,
\nWe can conclude that the mistake did by Kavi is the miswriting of eq (1)<\/p>\n

Do You Know How?<\/strong>
\nIn 4-6, solve each system using substitution.
\nQuestion 4.
\ny = \\(\\frac{1}{2}\\)x + 4
\nx – y = 8
\nAnswer:
\nThe given system of equations are:
\ny = \\(\\frac{1}{2}\\)x + 4 —– (1)
\nx – y = 8 —– (2)
\nNow,
\nFrom eq (2),
\ny = x – 8
\nSubstitute y in eq (1)
\nSo,
\nx – 8 = \\(\\frac{1}{2}\\)x + 4
\nx – \\(\\frac{1}{2}\\)x = 4 + 8
\n\\(\\frac{1}{2}\\)x = 12
\nx = 12 (2)
\nx = 24
\nSo,
\ny = x – 8
\ny = 24 – 8
\ny = 16
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (24, 16)<\/p>\n

Question 5.
\n3.25x \u2013 1.5y = 1.25
\n13x – y = 10
\nAnswer:
\nThe given system of equations are:
\n3.25x – 1.5y = 1.25 —- (1)
\n13x – y = 10 —-(2)
\nNow,
\nFrom eq (2),
\ny = 13x – 10
\nNow,
\nSubstitute y in eq (1)
\nSo,
\n3.25x – 1.5 (13x – 10) = 1.25
\n3.25x – 1.5 (13x) + 1.5(10) = 1.25
\n3.25x – 19.5x + 15 = 1.25
\n-16.25x = -13.75
\n16.25x = 13.75
\nx = 0.84
\nSo,
\ny = 13x – 10
\ny = 13 (0.84) – 10
\ny = 0.92
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (0.84, 0.92)<\/p>\n

Question 6.
\ny – 0.8x = 0.5
\n5y – 2.5 = 4x
\nAnswer:
\nThe given system of equations are:
\ny – 0.8x = 0.5 —- (1)
\n5y – 2.5 = 4x —— (2)
\nNow,
\nFrom eq (1),
\ny = 0.8x + 0.5
\nNow,
\nSubstitute y in eq (2)
\nSo,
\n5 (0.8x + 0.5) – 2.5 = 4x
\n5 (0.8x) + 5 (0.5) – 2.5 = 4x
\n4x + 2.5 – 2.5 = 4x
\n4x – 4x + 2.5 = 2.5
\n2.5 = 2.5
\nHence, from the above,
\nWe can conclude that the given system of equations has infinitely many solutions<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice In 7-9, solve the systems of equations.
\nQuestion 7.
\nPedro has 276 more hits than Ricky. Use substitution to solve the system of equations to find how many hits Pedro, p, and Ricky, r, have each recorded.
\np + r = 2,666
\np = r + 276
\n\"Envision
\nIt is given that Pedro has 276 more hits than Ricky
\nNow,
\nThe given system of equations are:
\np + r = 2,666
\np = r + 276
\nNow,
\nSTEP 1 Substitute for p to solve for r.
\np + r = 2,666
\nr + 276 + r = 2,666
\n2r + 276 = 2,666
\n2r = 2,390
\nr = 1,195
\nSTEP 2 Substitute for r to solve for p.
\np = r + 276
\np = 1,195 + 276
\np = 1,471
\nHence, from the above,
\nWe can conclude that
\nPedro has 1,471 hits, and Ricky has 1,195 hits.<\/p>\n

Question 8.
\n2y + 4.4x = -5
\ny = -2.2x + 4.5
\n2 (-2.2x + 4.5) + 4.4x = -5
\n-4.4x + 9 + 4.4x = -5
\n9 = -5
\nHence, from the above,
\nWe can conclude that
\nThe statement is not true. So, there is no solution.<\/p>\n

Question 9.
\nx + 5y = 0
\n25y = -5x
\nx = -5y
\n25y = -5 (-5y)
\n25y = 25y
\nHence, from the above,
\nWe can conclude that
\nThe statement is true. So, there are infinitely many solutions<\/p>\n

Question 10.
\nOn a certain hot summer day, 481 people used the public swimming pool. The daily prices are $1.25 for children and $2.25 for adults. The receipts for admission totaled $865.25. How many children and how many adults swam at the public pool that day?
\nAnswer:
\nIt is given that
\nOn a certain hot summer day, 481 people used the public swimming pool. The daily prices are $1.25 for children and $2.25 for adults. The receipts for admission totaled $865.25.
\nNow,
\nLet x be the number of children
\nLet y be the number of adults
\nSo,
\nx + y = 481 —– (1) [The number of people that used the public swimming pool]
\n1.25x + 2.25y = 865.25 —– (2) [The receipts for admission]
\nNow,
\nFrom eq (1),
\ny = 481 – x
\nSubstitute y in eq (2)
\nSo,
\n1.25x + 2.25 (481 – x) = 865.25
\n1.25x + 2.25 (481) – 2.25 (x) = 865.25
\n1.25x + 1,082.25 – 2.25x = 865.25
\n-x = -217
\nx = 217
\nNow,
\ny = 481 – x
\ny = 481 – 217
\ny = 264
\nHence,from the above,
\nWe can conclude that
\nThe number of children that swam at the pool is: 217
\nThe number of adults that swam at the pool is: 264<\/p>\n

Question 11.
\nConstruct Arguments<\/strong> Tim incorrectly says that the solution of the system of equations is x = -9, y = -4.
\n6x – 2y = -6
\n11 = y – 5x
\na. What is the correct solution?
\nAnswer:
\nThe given system of equations are:
\n6x – 2y = -6 —– (1)
\n11 = y – 5x —— (2)
\nNow,
\nFrom eq (2),
\ny = 5x + 11
\nSubstitute y in eq (1)
\nSo,
\n6x – 2 (5x + 11) = -6
\n6x – 2 (5x) – 2 (11) = -6
\n6x – 10x – 22 = -6
\n-4x = 16
\nx = –\\(\\frac{16}{4}\\)
\nx = -4
\nNow,
\ny = 5x + 11
\ny = 5 (-4) + 11
\ny = -20 + 11
\ny = -9
\nHence, from the above,
\nWe can conclude that the correct solution for the given system of equations is: (-4, -9)<\/p>\n

b. What error might Tim have made?
\nAnswer:
\nIt is given that
\nTim incorrectly says that the solution of the system of equations is x = -9, y = -4.
\nBut, from part (a),
\nWe can observe that x = -4, y = -9
\nHence, from the above,
\nWe can conclude that the error that Tim has made is the reversal of the values of x and y in the solution<\/p>\n

Question 12.
\nThe number of water bottles, y, filled in x minutes by each of two machines is given by the equations below. Use substitution to determine if there is a point at which the machines will have filled the same number of bottles.
\n160x + 2y = 50
\ny + 80x = 50
\nAnswer:
\nThe given system of equations are:
\n160x + 2y = 50 —— (1)
\ny + 80x = 50 ——- (2)
\nNow,
\nFrom eq (2),
\ny = 50 – 80x
\nNow,
\nSubstitute y in eq (1)
\nSo,
\n160x + 2 (50 – 80x) = 50
\n160x + 2 (50) – 2 (80x) = 50
\n160x + 100 – 160x = 50
\n100 = 50
\nHence, from the above,
\nWe can conclude that at any point, the machines will not have filled the same number of bottles<\/p>\n

Question 13.
\na. Use substitution to solve the system below.
\nx = 8y – 4
\nx + 8y = 6
\nAnswer:
\nThe given system of equations are:
\nx = 8y – 4 —– (1)
\nx + 8y = 6 —– (2)
\nNow,
\nSubstitute eq (1) in eq (2)
\nSo,
\n8y – 4 + 8y = 6
\n16y – 4 = 6
\n16y = 10
\ny = \\(\\frac{16}{10}\\)
\ny = 1.6
\nNow,
\nx = 8y – 4
\nx = 8 (1.6) – 4
\nx = 12.8 – 4
\nx = 8.8
\nHence, from the above,
\nWe can conclude that<\/p>\n

b. Reasoning<\/strong> Which expression would be easier to substitute into the other equation in order to solve the problem? Explain.
\nAnswer:
\nFrom part (a),
\nThe given system of equations are:
\nx = 8y – 4 —– (1)
\nx + 8y = 6 —– (2)
\nHence,
\nFrom the given equations,
\nIn order to reduce the number of steps,
\nEq (1) would be easier to substitute into the other equation in order to solve the problem<\/p>\n

Question 14.
\nThe perimeter of a frame is 36 inches. The length is 2 inches greater than the width. What are the dimensions of the frame?
\n\"Envision
\nAnswer:
\nThe given frame is:
\n\"Envision
\nFrom the given frame,
\nWe can observe that it is in the form of a rectangle
\nNow,
\nWe know that,
\nThe perimeter of a rectangle = 2 (Length + Width)
\nSo,
\n36 = 2 (W + 2 + W)
\n36 = 2 (2W + 2)
\n2W + 2 = \\(\\frac{36}{2}\\)
\n2W + 2 = 18
\n2W = 18 – 2
\n2W = 16
\nW = \\(\\frac{16}{2}\\)
\nW = 8
\nSo,
\nL = W + 2
\nL = 8 + 2
\nL = 10
\nHence, from the above,
\nWe can conclude that the dimensions of the frame are:
\nThe length of the frame is: 10 inches
\nThe width of the frame is: 8 inches<\/p>\n

Question 15.
\nHigher-Order Thinking<\/strong> The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds 500 people. By selling tickets, the members would like to raise $2,050 every night to cover all expenses. Let d represent the number of adult tickets sold at $6.50. Let s represent the number of student tickets sold at $3.50 each.
\na. If all 500 seats are filled for a performance, how many of each type of ticket must have been sold for the members to raise exactly $2,050?
\nAnswer:
\nIt is given that
\nThe members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds 500 people. By selling tickets, the members would like to raise $2,050 every night to cover all expenses. Let d represent the number of adult tickets sold at $6.50. Let s represent the number of student tickets sold at $3.50 each.
\nNow,
\nd + s = 500 —– (1) [The total number of people]
\n6.50d + 3.50s = 2,050 —– (2) [The total expenses]
\nSo,
\nFrom eq (1),
\nd = 500 – s
\nNow,
\nSubstitute d in eq (1)
\nSo,
\n6.50 (500 – s) + 3.50s = 2,050
\n6.50 (500) – 6.50s + 3.50s = 2,050
\n3,250 – 3s = 2,050
\n-3s =-1,200
\n3s = 1,200
\ns = \\(\\frac{1,200}{3}\\)
\ns = 400
\nSo,
\nd = 500 – s
\nd = 500 – 400
\nd = 100
\nHence, from the above,
\nWe can conclude that
\nThe number of student tickets is: 400
\nThe number of adult tickets is: 100<\/p>\n

b. At one performance there were three times as many student tickets sold as adult tickets. If there were 480 tickets sold at that performance, how much below the goal of $2,050 did ticket sales fall?
\nAnswer:
\nIt is given that there were three times as many student tickets as adult tickets
\nSo,
\ns = 3d
\nNow,
\nd + s = 480 —– (1)
\nSo,
\nd + 3d = 480
\n4d = 480
\nd = \\(\\frac{480}{4}\\)
\nd = 120
\nSo,
\ns = 3 (120)
\ns = 360
\nNow,
\nThe equation for the total expenses is:
\n6.50d + 3.50s = 6.50 (120) + 3.50 (360)
\n= 780 + 1,260
\n= 2,040
\nNow,
\nThe fall in the ticket sales = $2,050 – $2,040
\n= $10
\nHence, from the above,
\nWe can conclude that there is a fall of $10 to reach the goal of ticket sales of $2,050<\/p>\n

Assessment Practice<\/strong>
\nQuestion 16.
\nWhat statements are true about the solution of the system?
\ny = 145 – 5x
\n0.1y + 0.5x = 14.5
\n\u2610 There are infinitely many solutions.
\n\u2610 (20, 45) is a solution.
\n\u2610 (10, 95) is a solution.
\n\u2610 There is no solution.
\n\u2610 There is more than one solution.
\nAnswer:
\nLet the given options be named as A, B, C, D, and E
\nNow,
\nThe given system of equations are:
\ny = 145 – 5x —- (1)
\n0.1y + 0.5x = 14.5 —- (2)
\nNow,
\nSubstitute eq (1) in eq (2)
\nSo,
\n0.1 (145 – 5x) + 0.5x = 14.5
\n0.1 (145) – 0.1 (5x) + 0.5x = 14.5
\n14.5 – 0.5x + 0.5x = 14.5
\n14.5 – 0.5x = 14.5 – 0.5x
\nSo,
\nWe can say that there are infinitely many solutions for the given system of equations
\nHence, from the above,
\nWe can conclude that option A matches with the solution for the given system of equations<\/p>\n

Question 17.
\nAt an animal shelter, the number of dog adoptions one weekend was 10 less than 3 times the number of cat adoptions. The number of cat adoptions plus twice the number of dog adoptions was 8. How many cats and how many dogs were adopted that weekend?
\nAnswer:
\nIt is given that
\nAt an animal shelter, the number of dog adoptions one weekend was 10 less than 3 times the number of cat adoptions. The number of cat adoptions plus twice the number of dog adoptions was 8
\nNow,
\nLet the cat adoptions be x
\nLet the dog adoptions be y
\nSo,
\ny = 3x – 10 —– (1)
\nx + 2y = 8 ——- (2)
\nNow,
\nSubstitute eq (1) in eq (2)
\nSo,
\nx + 2 (3x – 10) = 8
\nx – 2 (10) + 2 (3x) = 8
\nx – 20 + 6x = 8
\n7x = 28
\nx = \\(\\frac{28}{4}\\)
\nx = 7
\nNow,
\ny = 3x – 10
\ny = 3 (7) – 10
\ny = 21 – 10
\ny = 11
\nHence, from the above,
\nWe can conclude that
\nThe number of dogs adopted that weekend is: 11
\nThe number of cats adopted that weekend is: 7<\/p>\n

Lesson 5.4 Solve Systems by Elimination<\/h3>\n

Solve & Discuss It!<\/strong>
\nA list of expressions is written on the board. How can you make a list of fewer expressions that has the same combined value as those shown on the board? Write the expressions and explain your reasoning.
\n\"Envision
\nI can… solve systems of equations using elimination.
\nAnswer:<\/p>\n

Look for Relationships
\nHow can you use what you know about combining like terms to make your list?
\nAnswer:
\nLike terms\u00a0are mathematical\u00a0terms\u00a0that have the exact same variables and exponents, but they can have different coefficients.\u00a0Combining like terms\u00a0will simplify a math problem and is also the proper form for writing a polynomial. To\u00a0combine like terms, just add the coefficients of each\u00a0like term<\/p>\n

Focus on math practices
\nReasoning Two expressions have a sum of 0. What must be true of the expressions?
\nAnswer:
\nIt is given that two expressions have a sum of 0
\nHence,
\nFrom the given information,
\nWe can conclude that the expressions are the same but they are of the opposite sign<\/p>\n

Essential Question<\/strong>
\nHow are the properties of equality used to solve systems of linear equations?
\nAnswer:
\nBasically, you can add or subtract anything you wish to both sides of any inequality, and you can multiply or divide, too, by any positive number, to get a simpler inequality, on the way to\u00a0solving\u00a0for the variable<\/p>\n

Try It!<\/strong><\/p>\n

Use elimination to solve the system of equations.
\n2r + 3s = 14
\n6r – 3s = 6
\nThe solution is r = ______, s = _____.
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\n2r + 3s = 14 —- (1)
\n6r – s = 6 —– (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the solutions for the given system of equations are:
\nr = \\(\\frac{25}{4}\\) and s = \\(\\frac{1}{2}\\)<\/p>\n

Convince Me!
\nWhat must be true about a system of equations for a term to be eliminated by adding or subtracting?
\nAnswer:
\nIn the\u00a0elimination method, you either\u00a0add or subtract\u00a0the\u00a0equations\u00a0to get an\u00a0equation\u00a0in one variable. When the coefficients of one variable are opposites you\u00a0add\u00a0the\u00a0equations\u00a0to\u00a0eliminate\u00a0a variable and when the coefficients of one variable are equal you\u00a0subtract\u00a0the\u00a0equations\u00a0to\u00a0eliminate\u00a0a variable.<\/p>\n

Try It!<\/strong><\/p>\n

Use elimination to solve the system of equations.
\n3x – 5y = -9
\nx + 2y = 8
\nAnswer:
\nThe given system of equations are:
\n3x – 5y = -9
\nx + 2y = 8
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (2, 3)<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

You can apply the properties of equality to solve systems of linear equations algebraically by eliminating a variable. Elimination is an efficient method when:
\n\u2022 like variable terms have the same or opposite coefficients.
\n\u2022 one or both equations can be multiplied so that like variable terms have the same or opposite coefficients.<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question<\/strong> How are the properties of equality used to solve systems of linear equations?
\nAnswer:
\nBasically, you can add or subtract anything you wish to both sides of any inequality, and you can multiply or divide, too, by any positive number, to get a simpler inequality, on the way to\u00a0solving\u00a0for the variable<\/p>\n

Question 2.
\nHow is solving a system of equations algebraically similar to solving the system by graphing? How is it different?
\nAnswer:
\nTo solve a system of linear equations graphically we graph both equations in the same coordinate system. The solution to the system will be at the point where the two lines intersect.<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Consider the system of equations. Would you solve this system by substitution or by elimination? Explain.
\n1\\(\\frac{3}{4}\\)x + y = 2\\(\\frac{3}{16}\\)
\n\\(\\frac{1}{4}\\)x – y = -1\\(\\frac{11}{16}\\)
\nAnswer:
\nThe given system of equations are:
\n1\\(\\frac{3}{4}\\)x + y = 2\\(\\frac{3}{16}\\)
\n\\(\\frac{1}{4}\\)x – y = -1\\(\\frac{11}{16}\\)
\nNow,
\nFrom the above equations,
\nWe can observe that we can eliminate y easily by using the elimination method
\nHence, from the above,
\nWe can conclude that we would solve the given system of equations by using the elimination method<\/p>\n

Do You Know How?<\/strong>
\nIn 4\u20136, solve each system of equations by using elimination.
\nQuestion 4.
\ny – x = 28
\ny + x = 156
\nAnswer:
\nThe given system of equations are:
\ny – x = 28 —- (1)
\ny + x = 156 —- (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (64, 92)<\/p>\n

Question 5.
\n3c + 6d = 18
\n6C – 2d = 22
\nAnswer:
\nThe given system of equations are:
\n3c + 6d = 18 — (1)
\n6c – 2d = 22 —– (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (4, 1)<\/p>\n

Question 6.
\n7x + 14y = 28
\n5x + 10y = 20
\nAnswer:
\nThe given system of equations are:
\n7x + 14y = 28 —– (1)
\n5x + 10y = 20 —– (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that there are infinitely many solutions for the given system of equations<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nLeveled Practice Solve the system of equations using elimination.
\n2x – 2y = 4
\n2x + y = 11
\nMultiply the first equation by ______.
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\n2x – 2y = 4 —– (1)
\n2x + y = 11 —– (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (\\(\\frac{13}{3}\\), \\(\\frac{7}{3}\\))<\/p>\n

Question 8.
\nSolve the system of equations using elimination.
\n2y – 5x = -2
\n3y + 2x = 35
\nAnswer:
\nThe given system of equations are:
\n2y – 5x = -2 —- (1)
\n3y + 2x = 35 —- (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (4, 9)<\/p>\n

Question 9.
\nIf you add Natalie’s age and Frankie’s age, the result is 44. If you add Frankie’s age to 3 times Natalie’s age, the result is 70. Write and solve a system of equations using elimination to find their ages.
\nAnswer:
\nIt is given that
\nIf you add Natalie’s age and Frankie’s age, the result is 44. If you add Frankie’s age to 3 times Natalie’s age, the result is 70.
\nNow,
\nLet Natalie’s age be n
\nLet Frankie’s age be f
\nSo,
\nFrom the given information,
\nThe system of equations that can be formed is:
\nn + f = 44 —- (1)
\nf + 3n = 70 —- (2)
\nNow,
\nEq (1) – Eq (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe age of Natalie is: 13 years
\nThe age of Frankie is: 31 years<\/p>\n

Question 10.
\nIf possible, use elimination to solve the system of equations.
\n5x + 10y = 7
\n4x + 8y = 3
\nAnswer:
\nThe given system of equations are:
\n5x + 10y = 7 —- (1)
\n4x + 8y = 3 —— (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the given system of equations has no solution<\/p>\n

Question 11.
\nAt a basketball game, a team made 56 successful shots. They were a combination of 1- and 2-point shots. The team scored 94 points in all. Use elimination to solve the system of equations to find the number of each type of shot.
\nx + y = 56
\nx + 2y = 94
\nAnswer:
\nIt is given that
\nAt a basketball game, a team made 56 successful shots. They were a combination of 1- and 2-point shots. The team scored 94 points in all
\nNow,
\nThe given system of equations that represent the given situation is:
\nx + y = 56 —- (1)
\nx + 2y = 94 —- (2)
\nNow,
\nEq (1) – Eq (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of 1-point shots is: 18
\nThe number of 2-point shots is: 38<\/p>\n

Question 12.
\nTwo trains, Train A and Train B, weigh a total of 312 tons. Train A is heavier than Train B. The difference in their weights is 170 tons. Use elimination to solve the system of equations to find the weight of each train.
\na + b = 312
\na – b= 170
\n\"Envision
\nAnswer:
\nIt is given that
\nTwo trains, Train A and Train B weigh a total of 312 tons. Train A is heavier than Train B. The difference in their weights is 170 tons.
\nNow,
\nThe system of equations given that describes the given situation is:
\na + b = 312 —– (1)
\na – b = 170 —— (2)
\nNow,
\nEq (1) – Eq (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe weight of Train A is: 241 tons
\nThe weight of Train B is: 71 tons<\/p>\n

Question 13.
\nA deli offers two platters of sandwiches. Platter A has 2 roast beef sandwiches and 3 turkey sandwiches. Platter B has 3 roast beef sandwiches and 2 turkey sandwiches.
\n\"Envision
\na. Model with Math Write a system of equations to represent the situation.
\nAnswer:
\nIt is given that
\nA deli offers two platters of sandwiches. Platter A has 2 roast beef sandwiches and 3 turkey sandwiches. Platter B has 3 roast beef sandwiches and 2 turkey sandwiches.
\nNow,
\nLet a piece of roast beef sandwich be r
\nLet a piece of turkey beef sandwich be t
\nSo,
\nIn plate A,
\nThe number of sandwiches is: 2r + 3t
\nIn plate B,
\nThe number of sandwiches is: 3r + 2t
\nNow,
\nFrom the above figure,
\nThe total cost of plates A and B are:
\n2r + 3t = 31
\n3r + 2t = 29
\nHence, from the above,
\nWe can conclude that the system of equations that represent the given situation is:
\n2r + 3t = 31
\n3r + 2t = 29<\/p>\n

b. What is the cost of each sandwich?
\nAnswer:
\nFrom part (a),
\nThe system of equations are:
\n2r + 3t = 31 —– (1)
\n3r + 2t = 29 —– (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe cost of each roast beef sandwich is: $5
\nThe cost of each turkey beef sandwich is: $7<\/p>\n

Question 14.
\nConsider the system of equations.
\nx – 3.1y = 11.5
\n-x + 3.5y = -13.5
\na. Solve the system by elimination.
\nAnswer:
\nThe given system of equations are:
\nx – 3.1y = 11.5 —- (1)
\n-x + 3.5y = -13.5 —— (2)
\nNow,
\nEq (1) – Eq (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (-4, -5)<\/p>\n

b. If you solved this equation by substitution instead, what would the solution be? Explain.
\nAnswer:
\nBy using the substitution method,
\nThe solution for the given system of equations is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the substitution method is the same as the solution for the elimination method<\/p>\n

Question 15.
\nHigher-Order Thinking<\/strong> Determine the number of solutions for this system of equations by inspection only. Explain.
\n3x + 4y = 17
\n21x + 28y = 109
\nAnswer:
\nThe given system of equations are:
\n3x + 4y = 17 —– (1)
\n21x + 28y = 109 —— (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that there is no solution for the given system of equations<\/p>\n

Assessment Practice<\/strong>
\nQuestion 16.
\nFour times a number r plus half a number s equals 12. Twice the number r plus one fourth of the number s equals 8. What are the two numbers?
\nAnswer:
\nIt is given that
\nFour times a number r plus half a number s equals 12. Twice the number r plus one-fourth of the number s equals 8.
\nSo,
\n4r + \\(\\frac{1}{2}\\)s = 12 —- (1)
\n2r + \\(\\frac{1}{4}\\)s = 8 —- (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the values for the 2 numbers are not possible<\/p>\n

Question 17.
\nSolve the system of equations.
\n3m + 3n = 36
\n8m – 5n = 31
\nAnswer:
\nThe given system of equations are:
\n3m + 3n = 36 —- (1)
\n8m – 5n = 31 —– (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (7, 5)<\/p>\n

3-ACT MATH<\/strong><\/p>\n

3-Act Mathematical Modeling: Ups and Downs
\n\"Envision<\/p>\n

ACT 1<\/strong>
\nQuestion 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:<\/p>\n

\"Envision
\nQuestion 2.
\nWrite the Main Question you will answer.
\nAnswer:<\/p>\n

Question 3.
\nMake a prediction to answer this Main Question.
\nThe person who wins took the ______.
\nAnswer:<\/p>\n

Question 4.
\nConstruct Arguments Explain how you arrived at your prediction.
\nAnswer:<\/p>\n

ACT 2<\/strong>
\nQuestion 5.
\nWhat information in this situation would be helpful to know? How would you use that information?
\nAnswer:<\/p>\n

Question 6.
\nUse Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
\nAnswer:<\/p>\n

Question 7.
\nModel with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 8.
\nWhat is your answer to the Main Question? Does it differ from your prediction? Explain.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong>
\nQuestion 9.
\nWrite the answer you saw in the video.
\n\"Envision
\nAnswer:<\/p>\n

Question 10.
\nReasoning<\/strong> Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\nAnswer:<\/p>\n

Question 11.
\nMake Sense and Persevere Would you change your model now that you know the answer? Explain.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3 Extension<\/strong>
\nReflect
\nQuestion 12.
\nModel with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\n\"Envision
\nAnswer:<\/p>\n

Question 13.
\nReason Abstractly A classmate solved the problem using equations with independent variable a and dependent variable b. What do these variables represent in the situation?
\nAnswer:<\/p>\n

SEQUEL<\/strong>
\nQuestion 14.
\nGeneralize Write an equation or inequality to represent all numbers of flights for which the elevator is faster.
\n\"Envision
\nAnswer:<\/p>\n

Topic 5 REVIEW<\/h3>\n

Topic Essential Question<\/strong>
\nWhat does it mean to solve a system of linear equations?
\nAnswer:
\nThe\u00a0solution to a system of linear equations is the point at which the lines representing the linear equations intersect. Two lines in the XY -plane can\u00a0intersect once, never intersect, or completely overlap.<\/p>\n

Vocabulary Review<\/strong><\/p>\n

Complete each definition and then provide an example of each vocabulary word.
\nVocabulary<\/strong>
\nsolution of a system of linear equations
\nsystem of linear equations
\n\"Envision
\nAnswer:
\n\"\"<\/p>\n

Use Vocabulary in Writing
\nDescribe how you can find the number of solutions of two or more equations by using the slope and the y-intercept. Use vocabulary terms in your description.
\nAnswer:
\nWhen slopes and y-intercepts are different, the two lines intersect at one point,
\nThere is only 1 solution.
\nWhen\u00a0slopes\u00a0are the same, the lines may be either parallel (different\u00a0y-intercepts)
\nThere are no solutions
\nIf the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
\nSo, there are infinitely many solutions<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

Lesson 5.1 Estimate Solutions by Inspection<\/strong><\/p>\n

Quick Review<\/strong>
\nThe slopes and y-intercepts of the linear equations in a system determine the relationship between the lines and the number of solutions.
\n\"Envision<\/p>\n

Example<\/strong>
\nHow many solutions does the system of equations have? Explain.
\nAnswer:
\ny + 2x = 6
\ny – 8 = -2x
\nWrite each equation in slope-intercept form.
\ny = -2x + 6
\ny = -2x + 8
\nIdentify the slope and y-intercept of each equation.
\nFor the equation, y = -2x + 6, the slope is \u20132 and the y-intercept is 6.
\nFor the equation, y = -2x + 8, the slope is -2 and the y-intercept is 8.
\nThe equations have the same slope but different y-intercepts, so the system has no solution.<\/p>\n

Practice<\/strong>
\nDetermine whether the system of equations has one solution, no solution, or infinitely many solutions.
\nQuestion 1.
\ny \u2013 13 = 5x
\ny \u2013 5x = 12
\nAnswer:
\nThe given system of equations are:
\ny – 13 = 5x
\ny – 5x = 12
\nSo,
\nThe required system of equations are:
\ny = 5x + 13 —– (1)
\ny = 5x + 12 —– (2)
\nNow,
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = 5, b = 13
\nFor the second equation,
\nm = 5, b = 12
\nWe know that,
\nWhen\u00a0slopes\u00a0are the same, the lines may be either parallel (different\u00a0y-intercepts)
\nThere are no solutions
\nHence, from the above,
\nWe can conclude that there is no solution for the given system of equations<\/p>\n

Question 2.
\ny = 2x + 10
\n3y \u2013 6x = 30
\nAnswer:
\nThe given system of equations are:
\ny = 2x + 10 —– (1)
\n3y – 6x = 30 —– (2)
\nSo,
\nDivide eq (2) by 3
\ny – 2x = 10
\nSo,
\ny = 2x + 10 —– (3)
\nNow,
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFor the first equation,
\nm = 2, b = 10
\nFor the third equation,
\nm = 2, b = 10
\nWe know that,
\nIf the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
\nSo, there are infinitely many solutions
\nHence, from the above,
\nWe can conclude that there are infinitely many solutions for the given system of equations<\/p>\n

Question 3.
\n-3x + \\(\\frac{1}{3}\\)y = 12
\n2y = 18x + 72
\nAnswer:
\nThe given system of equations are:
\n-3x + \\(\\frac{1}{3}\\)y = 12 —– (1)
\n2y = 18x + 72 —- (2)
\nSo,
\nMultiply eq (1) with 3
\n-9x + y = 36
\ny = 9x + 36 —– (3)
\nNow,
\nDivide eq (2) with 2
\nSo,
\ny = 9x + 36 —– (4)
\nNow,
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFrom the third equation,
\nm = 9, b = 36
\nFrom the fourth equation,
\nm = 9, b = 36
\nWe know that,
\nIf the lines are the same line (Same slopes and same y-intercepts), they intersect at all points,
\nSo, there are infinitely many solutions
\nHence, from the above,
\nWe can conclude that there are infinitely many solutions for the given system of equations<\/p>\n

Question 4.
\ny – \\(\\frac{1}{4}\\)x = -1
\ny – 2 = 4x
\nAnswer:
\nThe given system of equations are:
\ny – \\(\\frac{1}{4}\\)x = -1 —- (1)
\ny – 2 = 4x —— (2)
\nNow,
\ny = \\(\\frac{1}{4}\\)x – 1 —- (3)
\ny = 4x + 2 —- (4)
\nCompare the above equations with
\ny = mx + b
\nSo,
\nFrom the third equation,
\nm = \\(\\frac{1}{4}\\), b = -1
\nFrom the fourth equation,
\nm = 4, b = 2
\nWe know that,
\nWhen slopes and y-intercepts are different, the two lines intersect at one point,
\nThere is only 1 solution.
\nHence, from the above,
\nWe can conclude that there is only 1 solution for the given system of equations<\/p>\n

Question 5.
\nMichael and Ashley each buy x pounds of turkey and y pounds of ham. Turkey costs $3 per pound at Store A and $4.50 per pound at Store B. Ham costs $4 per pound at Store A and $6 per pound at Store B. Michael spends $18 at Store A, and Ashley spends $27 at Store B. Could Michael and Ashley have bought the same amount of turkey and ham? Explain.
\nAnswer:
\nIt is given that
\nMichael and Ashley each buy x pounds of turkey and y pounds of ham. Turkey costs $3 per pound at Store A and $4.50 per pound at Store B. Ham costs $4 per pound at Store A and $6 per pound at Store B. Michael spends $18 at Store A, and Ashley spends $27 at Store B.
\nNow,
\nTake the number of pounds for the turkey to be x and that for the ham to be y
\nFor store A where Michael spent $18,
\nTurkey cost $3 per pound —- 3x
\nHam cost $4 per pound——4y
\nSo,
\nThe equation for cost will be;
\n3x + 4y = 18
\nNow,
\nFor store B where Ashley spent $27
\nTurkey cost $4.5 per pound
\nHam cost $6 per pound
\nSo,
\nThe equation for cost is:
\n4.5x + 6y = 27
\nSo,
\nThe two equations are;
\n3x + 4y = 18 —— (1)
\n4.5x + 6y = 27 —– (2)
\nNow,
\nDivide eq (2) with 3
\nSo,
\n1.5x + 2y = 9 —- (3)
\nMultiply the above equation with 2
\nSo,
\n3x + 4y = 18 — (4)
\nNow,
\nCompare eq (1) and eq (4) with
\ny = mx + b
\nSo,
\nFrom the eq (1),
\nm = 3, b = 4
\nFrom the eq (2),
\nm = 3, b = 4
\nFrom the above,
\nWe can observe that the slopes and the y-intercepts are equal
\nSo,
\nBoth the equations are in the same line
\nHence, from the above,
\nWe can conclude that Michael and Ashley bought the same amount of Turkey and Ham<\/p>\n

Lesson 5.2 Solve Systems by Graphing<\/strong><\/p>\n

Quick Review<\/strong>
\nSystems of equations can be solved by looking at their graphs. A system with one solution has one point of intersection. A system with infinitely many solutions has infinite points of intersection. A system with no solution has no points of intersection.<\/p>\n

Example<\/strong>
\nGraph the system and determine its solution.
\ny = x + 4
\ny = -2x + 1
\nAnswer:
\nGraph each equation in the system on the same coordinate plane.
\n\"Envision
\nThe point of intersection is (-1, 3). This means the solution to the system is (-1, 3).<\/p>\n

Practice<\/strong>
\nGraph each system and find the solution(s).
\nQuestion 1.
\ny = \\(\\frac{1}{2}\\)x + 1
\n-2x + 4y = 4
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\ny = \\(\\frac{1}{2}\\)x + 1
\n-2x + 4y = 4
\nSo,
\nThe representation of the given system of equations in a coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the given system of equations has infinitely many solutions<\/p>\n

Question 2.
\ny = -x – 3
\ny + x = 2
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\ny = -x – 3
\ny + x = 2
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that there is no solution for the given system of equations<\/p>\n

Question 3.
\n2y = 6x + 4
\ny = -2x + 2
\n\"Envision
\nAnswer:
\nThe given system of equations are:
\n2y = 6x + 4
\ny = -2x + 2
\nSo,
\nThe representation of the given system of equations in the coordinate plane is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (0, 2)<\/p>\n

Lesson 5.3 Solve Systems by Substitution<\/strong><\/p>\n

Quick Review<\/strong>
\nTo solve a system by substitution, write one equation for a variable in terms of the other. Substitute the expression into the other equation and solve. If the result is false, the system has no solution. If true, it has infinitely many solutions. If the result is a value, substitute to solve for the other variable.<\/p>\n

Example<\/strong>
\nUse substitution to solve the system.
\ny = x + 1
\ny = 5x \u2013 3
\nAnswer:
\nSubstitute x + 1 for y in the second equation.
\n(x + 1) = 5x – 3
\n4 = 4x
\n1 = x
\nSubstitute 1 for x in the first equation.
\ny = (1) + 1 = 2
\nThe solution is x = 1, y = 2.<\/p>\n

Practice<\/strong>
\nUse substitution to solve each system.
\nQuestion 1.
\n-3y = -2x \u2013 1
\ny = x – 1
\nAnswer:
\nThe given system of equations are:
\n-3y = -2x – 1 —– (1)
\ny = x – 1 —- (2)
\nNow,
\nSubstitute eq (2) in eq (1)
\nSo,
\n-3 (x – 1) = -2x – 1
\n-3 (x) + 3 (1) = -2x – 1
\n-3x + 3 = -2x – 1
\n-3x + 2x = -1 – 3
\n-x = -4
\nx = 4
\nSo,
\ny = x – 1
\ny = 4 – 1
\ny = 3
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (4, 3)<\/p>\n

Question 2.
\ny = 5x + 2
\n2y \u2013 4 = 10x
\nAnswer:
\nThe given system of equations are:
\ny = 5x + 2 —- (1)
\n2y – 4 = 10 —– (2)
\nNow,
\nSubstitute eq (1) in eq (2)
\nSo,
\n2 (5x + 2) – 4 = 10
\n2 (5x) + 2 (2) – 4 = 10
\n10x = 10
\nx = 1
\nSo,
\ny = 5x + 2
\ny = 5 + 2
\ny = 7
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (1, 7)<\/p>\n

Question 3.
\n2y – 8 = 6x
\ny = 3x + 2
\nAnswer:
\nThe given system of equations are:
\n2y – 8 = 6x —- (1)
\ny = 3x + 2 —– (2)
\nNow,
\nSubstitute eq (2) in eq (1)
\nSo,
\n2 (3x + 2) – 8 = 6x
\n2 (3x) + 2 (2) – 8 = 6x
\n6x + 4 – 8 = 6x
\n4 = 8
\nHence, from the above,
\nWe can conclude that there is no solution for the given system of equations<\/p>\n

Question 4.
\n2y \u2013 2 = 4x
\ny = -x + 4
\nAnswer:
\nThe given system of equations are:
\n2y – 2 = 4x —– (1)
\ny = -x + 4 —– (2)
\nNow,
\nSubstitute eq (2) in eq (1)
\nSo,
\n2 (-x + 4) – 2 = 4x
\n2 (-x) + 2 (4) – 2 = 4x
\n-2x + 8 – 2 = 4x
\n4x + 2x = 6
\n6x = 6
\nx = 1
\nSo,
\ny = -x + 4
\ny = -1 + 4
\ny = 3
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (1, 3)<\/p>\n

Lesson 5.4 Solve Systems by Elimination<\/strong><\/p>\n

Quick Review<\/strong>
\nTo solve a system by elimination, multiply one or both equations to make opposite terms. Add (or subtract) the equations to eliminate one variable. Substitute to solve for the other variable.<\/p>\n

Example<\/strong>
\nUse elimination to solve the system.
\n2x – 9y = -5
\n4x – 6y = 2
\nMultiply the first equation by -2.Then add.
\n\"Envision
\ny = 1
\nSubstitute 1 for y in the first equation.
\n2x – 9(1) = -5
\n2x – 9 = -5
\n2x = 4
\nx = 2
\nThe solution is x = 2, y = 1.<\/p>\n

Practice<\/strong>
\nUse elimination to solve each system.
\nQuestion 1.
\n-2x + 2y = 2
\n4x \u2013 4y = 4
\nAnswer:
\nThe given system of equations are:
\n-2x + 2y = 2 —– (1)
\n4x – 4y = 4 ——- (2)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the given system of equations has no solution<\/p>\n

Question 2.
\n4x + 6y = 40
\n-2x + y = 4
\nAnswer:
\nThe given system of equations are:
\n4x + 6y = 40
\n-2x + y = 4
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the solution for the given system of equations is: (1, 6)<\/p>\n

Question 3.
\nA customer at a concession stand bought 2 boxes of popcorn and 3 drinks for $12. Another customer bought 3 boxes of popcorn and 5 drinks for $19. How much does a box of popcorn cost? How much does a drink cost?
\nAnswer:
\nIt is given that
\nA customer at a concession stands bought 2 boxes of popcorn and 3 drinks for $12. Another customer bought 3 boxes of popcorn and 5 drinks for $19
\nNow,
\nLet each box of popcorn be p
\nLet each drink be d
\nSo,
\nFor Customer A,
\n2p + 3d = $12 —– (1)
\nFor Customer B,
\n3p + 5d = $19 —– (2)
\nNow,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe cost of each popcorn box is: $3
\nThe cost of each drink is: $2<\/p>\n

Topic 6 Fluency Practice<\/h3>\n

Pathfinder<\/strong>
\nShade a path from START to FINISH. Follow the solutions to the equations from least to greatest. You can only move up, down, right, or left.
\nI can… solve multistep equations using the Distributive Property.
\n\"Envision<\/p>\n","protected":false},"excerpt":{"rendered":"

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