Learning ObjectivesBy the end of this section, you will be able to: Show
Be Prepared 5.4Before you get started, take this readiness quiz. Simplify −5(3−x)−5(3−x)
.
Be Prepared 5.5Simplify 4−2(n+5)4−2(n+5).
Be Prepared 5.6Solve for yy: 8y−8=32−2y8y−8=
32−2y
Be Prepared 5.7Solve for xx: 3x−9y=−33x−9y=−3
Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph. In this section, we will solve systems of linear equations by the substitution method. Solve a System of Equations by SubstitutionWe will use the same system we used first for graphing. {2x+y=7x−2y=6{2x+y=7x−2y=6 We will first solve one of the equations for either x or y. We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy. Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those! After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true. We’ll fill in all these steps now in Example 5.13.
Example 5.13How to Solve a System of Equations by SubstitutionSolve the system by substitution. {2x+y=7x−2 y=6{2x+y=7x−2y=6
Try It 5.25Solve the system by substitution. {−2x+y=−11x+3y=9{ −2x+y=−11x+3y=9
Try It 5.26Solve the system by substitution. {x+3y=104x+y=18{ x+3y=104x+y=18
How ToSolve a system of equations by substitution.
If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in Example 5.14.
Example 5.14Solve the system by substitution. {x+y=−1y=x+5 {x+y=−1y=x+5
Try It 5.27Solve the system by substitution. {x+y=6y=3x−2{x+y=6y=3x−2
Try It 5.28Solve the system by substitution. {2x−y=1y=−3x−6{ 2x−y=1y=−3x−6 If the equations are given in standard form, we’ll need to start by solving for one of the variables. In this next example, we’ll solve the first equation for y.
Example 5.15Solve the system by substitution. {3x+y=52x+4y=−10 {3x+y=52x+4y=−10
Try It 5.29Solve the system by substitution. {4x+y=23x+2y=−1{ 4x+y=23x+2y=−1
Try It 5.30Solve the system by substitution. {−x+y=44x−y=2{ −x+y=44x−y=2 In Example 5.15 it was easiest to solve for y in the first equation because it had a coefficient of 1. In Example 5.16 it will be easier to solve for x.
Example 5.16Solve the system by substitution. {x−2y=−23x+2y=34 {x−2y=−23x+2y=34
Try It 5.31Solve the system by substitution. {x−5y=134x−3y=1{ x−5y=134x−3y=1
Try It 5.32Solve the system by substitution. {x−6y=−62x−4y=4{ x−6y=−62x−4y=4 When both equations are already solved for the same variable, it is easy to substitute!
Example 5.17Solve the system by substitution. {y=−2x+5y=12x {y=−2x+5y=12x
Try It 5.33Solve the system by substitution. {y=3x−16y=13x{ y=3x−16y=13x
Try It 5.34Solve the system by substitution. {y=−x+10y=14x{ y=−x+10y=14x Be very careful with the signs in the next example.
Example 5.18Solve the system by substitution. {4x+2y=46x−y=8 {4x+2y=46x−y=8
Try It 5.35Solve the system by substitution. {x−4y=−4−3x+4y=0{ x−4y=−4−3x+4y=0
Try It 5.36Solve the system by substitution. {4x−y=02x−3y=5{ 4x−y=02x−3y=5 In Example 5.19, it will take a little more work to solve one equation for x or y.
Example 5.19Solve the system by substitution. {4x−3y=615y−20x=−30 {4x−3y=615y−20x=−30
Try It 5.37Solve the system by substitution. {2x−3y=12−12y+8x=48 {2x−3y=12−12y+8x=48
Try It 5.38Solve the system by substitution. {5x+2y=12−4y−10x=−24 {5x+2y=12−4y−10x=−24 Look back at the equations in Example 5.19. Is there any way to recognize that they are the same line? Let’s see what happens in the next example.
Example 5.20Solve the system by substitution. {5x−2y=−10y=52x {5x−2y=−10y=52x
Try It 5.39Solve the system by substitution. {3x+2y=9y=−32x+1{3x+2y=9y=−32x+1
Try It 5.40Solve the system by substitution. {5x−3y=2y=53x−4{5x−3y=2y=53x−4 Solve Applications of Systems of Equations by SubstitutionWe’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.
How ToHow to use a problem solving strategy for systems of linear equations.
Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?
Example 5.21The sum of two numbers is zero. One number is nine less than the other. Find the numbers.
Try It 5.41The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.
Try It 5.42The sum of two number is −6. One number is 10 less than the other. Find the numbers. In the Example 5.22, we’ll use the formula for the perimeter of a rectangle, P = 2L + 2W.
Example 5.22The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.
Try It 5.43The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.
Try It 5.44The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width of the rectangle. For Example 5.23 we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.
Example 5.23The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.
Try It 5.45The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.
Try It 5.46The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.
Example 5.24Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 plus $15 for each training session. Option B would pay her $10,000 + $40 for each training session. How many training sessions would make the salary options equal?
Try It 5.47Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. The second pays a salary of $20,000 plus a commission of $50 for each policy sold. How many policies would need to be sold to make the total pay the same?
Try It 5.48Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal? Section 5.2 ExercisesPractice Makes PerfectSolve a System of Equations by Substitution In the following exercises, solve the systems of equations by substitution. 71. {2x+y=−43x −2y=−6{2x+y=−43x−2y=−6 72. {2x+y=−23x−y=7{ 2x+y=−23x−y=7 73. {x−2y=−52x −3y=−4{x−2y=−52x−3y=−4 74. {x−3y=−92x+5y=4{x−3y=−92x+5y=4 75. {5x−2y=−6y =3x+3{5x−2y=−6y=3x+3 76. {−2x+2y=6y=−3x+1{−2x+2y=6y=−3x+1 77. {2x+3y=3y =−x+3{2x+3y=3y=−x+3 78. {2x+5y=−14y=−2x+2{2x+5y=−14y=−2x+2 79. {2x+5y=1y =13x−2{2x+5y=1y=13x−2 80. {3x+4y=1y=−25x+2{3x+4y=1y=−25x+2 81. {3x−2y=6y =23x+2{3x−2y=6y=23x+2 82. {−3x−5y=3y=12x−5{−3x−5y=3y=12x−5 83. {2x+y=10−x +y=−5{2x+y=10−x+y=−5 84. {−2x+y=10−x+2y=16{−2x+y =10−x+2y=16 85. {3x+y=1−4x +y=15{3x+y=1−4x+y=15 86. {x+y=02x+3y=−4{x+y=0 2x+3y=−4 87. {x+3y=13x+5y=−5{x+3y= 13x+5y=−5 88. {x+2y=−1 2x+3y=1{x+2y=−12x+3y=1 89. {2x+y=5 x−2y=−15{2x+y=5x−2y=−15 90. {4x+y=10x−2y=−20{ 4x+y=10x−2y=−20 91. {y=−2x−1y= −13x+4{y=−2x−1y=−13x+4 92. {y=x−6y=−32x+4 {y=x−6y=−32x+4 93. {y=2x−8y= 35x+6{y=2x−8y=35x+6 94. {y=−x−1y=x+7{y=−x−1y=x+7 95. {4x+2y=88 x−y=1{4x+2y=88x−y=1 96. {−x−12y=−12x−8y=−6{ −x−12y=−12x−8y=−6 97. {15x+2y=6−5 x+2y=−4{15x+2y=6−5x+2y=−4 98. {2x−15y=712x+2y=−4{ 2x−15y=712x+2y=−4 99. {y=3x6x−2 y=0{y=3x6x−2y=0 100. {x=2y4x−8y=0{x=2y4x−8y=0 101. {2x+16y=8−x−8y=−4{2x+16 y=8−x−8y=−4 102. {15x+4y =6−30x−8y=−12{15x+4y=6−30x−8y=−12 103. {y= −4x4x+y=1{y=−4x4x+y=1 104. {y=−14xx+4y=8{ y=−14xx+4y=8 105. {y=78x+4−7x+8y=6{y=78x+4−7x+8y=6 106. {y=−23x+52x+3y=11 {y=−23x+52x+3y=11 Solve Applications of Systems of Equations by Substitution In the following exercises, translate to a system of equations and solve. 107. The sum of two numbers is 15. One number is 3 less than the other. Find the numbers. 108. The sum of two numbers is 30. One number is 4 less than the other. Find the numbers. 109. The sum of two numbers is −26. One number is 12 less than the other. Find the numbers. 110. The perimeter of a rectangle is 50. The length is 5 more than the width. Find the length and width. 111. The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width. 112. The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width. 113. The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width. 114. The measure of one of the small angles of a right triangle is 14 more than 3 times the measure of the other small angle. Find the measure of both angles. 115. The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles. 116. The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles. 117. The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles. 118. Maxim has been offered positions by two car dealers. The first company pays a salary of $10,000 plus a commission of $1,000 for each car sold. The second pays a salary of $20,000 plus a commission of $500 for each car sold. How many cars would need to be sold to make the total pay the same? 119. Jackie has been offered positions by two cable companies. The first company pays a salary of $ 14,000 plus a commission of $100 for each cable package sold. The second pays a salary of $20,000 plus a commission of $25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same? 120. Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal? 121. Mitchell currently sells stoves for company A at a salary of $12,000 plus a $150 commission for each stove he sells. Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal? Everyday Math122. When Gloria spent 15 minutes on the elliptical trainer and then did circuit training for 30 minutes, her fitness app says she burned 435 calories. When she spent 30 minutes on the elliptical trainer and 40 minutes circuit training she burned 690 calories. Solve the system {15e+30c=43530e+40c=690 {15e+30c=43530e+40c=690 for ee , the number of calories she burns for each minute on the elliptical trainer, and cc, the number of calories she burns for each minute of circuit training. 123. Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system {56s=70ts=t+12{56s=70 ts=t+12.
Writing Exercises124. Solve the
system of equations ⓐ by graphing. ⓑ by substitution. ⓒ Which method do you prefer? Why? 125. Solve the system of equations Self Checkⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ After reviewing this checklist, what will you do to become confident for all objectives? How do you solve systems of equations by substitution?Here's how it goes:. Step 1: Solve one of the equations for one of the variables. Let's solve the first equation for y: ... . Step 2: Substitute that equation into the other equation, and solve for x. ... . Step 3: Substitute x = 4 x = 4 x=4 into one of the original equations, and solve for y.. How do you do substitution?The method of substitution involves three steps: Solve one equation for one of the variables. Substitute (plug-in) this expression into the other equation and solve. Resubstitute the value into the original equation to find the corresponding variable.
What is substituting method?The substitution method is the algebraic method to solve simultaneous linear equations. As the word says, in this method, the value of one variable from one equation is substituted in the other equation.
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