150 Lesson 4.4 ~ Solving Systems by Substitution

Solving a system of linear equations by graphing or input-output tables is convenient when the ordered pair solution contains only small integers. This will not occur with every system of equations. This lesson shows another method for solving a system of linear equations called the substitution method.

use the substitution method to solve the system of linear equations. y = −2x + 5 4x + 3y = 9 The first equation has an isolated variable. Since y = −2x + 5, substitute −2x + 5 for y in the second equation and solve for x.

4x + 3y = 9

4x + 3(−2x + 5) = 94x + −6x + 15 = 9−2x + 15 = 9 −15 −15 −2x ___ −2 = −6 __ −2 x = 3Substitute 3 for x in the first equation. y = −2(3) + 5 y = −6 + 5 y = −1Verify that (3, −1) makes both equations true. y = −2x + 5 4x + 3y = 9−1 ?= −2(3) + 5 4(3) + 3(−1) ?= 9−1 ?= −6 + 5 12 + −3 ?= 9 −1 = −1 9 = 9

ExamplE 1

solution

solving sysTEms by subsTiTuTion

Lesson 4.4

Lesson 4.4 ~ Solving Systems by Substitution 151

You can also graph the two linear equations to verify that your solution matches the point of intersection. Looking at this graph you can see why substitution wasa better method than graphing. It is hard to determine the exact point of intersection on the graph.

Zach lives in Salina, Kansas. Gina lives 45 miles east in Junction City, Kansas. Gina and Zach both leave their houses at the same time, heading east on I-70. Zach drives 65 miles per hour and Gina drives 55 miles per hour. Zach wants to determine how long it will take before he catches up with Gina.

step 1: Copy the equations listed below and identify which equation corresponds to Zach and which equation corresponds to Gina. The y-variable represents the distance from Zach’s house. Describe what the x-variable represents in this situation. y = 55x + 45 y = 65x

step 2: Solve this system of equations using the substitution method. step 3: How many hours will it take before Zach catches Gina? How far will Zach have driven at this point?

step 4: Explain how you know your answer is correct.

ExErcisEs

identify the equation that has an isolated variable. state which variable is isolated. 1. Equation #1 3x − 5y = 10 2. Equation #1 y = 1 _ 2 x − 4 Equation #2 x = 4 − 4y Equation #2 x − 5y = 5

3. Equation #1 −x + 3y = 10 4. Equation #1 12 − 6y = x Equation #2 y = 4 − 2x Equation #2 2x + y = 6

5. Nathan wants to know why he cannot always just use graphing to solve a system of linear equations. How would you answer his question? 6. Explain two different methods that can be used to verify an answer when solving a system of equations using substitution.

7. Isolate the y-variable in the equation 2x + y = 7.

8. Isolate the x-variable in the equation 3x = 6 + 9y.

ExplorE! a Trip on i-70

152 Lesson 4.4 ~ Solving Systems by Substitution

solve each system of equations using the substitution method. show all work necessary to justify your answer. 9. x = y − 3 10. 3x − y = 7 11. y = 10 − 2x 5x + 3y = 1 y = 2x − 4 3x − 2y = 22

12. x = 12 + 3y 13. y = 15 + x 14. 5y + 7x = 2 2x + 5y = −20 2x + 5y = 26 x = 2 + y

15. y = −5 + x 16. 2x + 3y = 17 17. x − 7y = 4 −2x + y = −4 2x + y = 3 3x + y = −10

18. Hank’s Ice Cream Shop sells single and double scoop cones. The single-scoop cones cost $2.00 and the double-scoop cones cost $2.50. In one day he sold 230 cones for a total of $498 in sales. a. Explain why the equations x + y = 230 and 2x + 2.5y = 498 represent this situation. b. What does x represent in the equations in part a? What does y represent? c. Isolate one variable in an equation. Solve the system of equations using the substitution method. d. What is the real-world meaning of the solution to the system?

19. Emma picked two numbers, x and y. She told her teacher that the sum of the two numbers was 46 and the difference of the two numbers was 12. a. Write two different linear equations that model what Emma told her teacher. b. Isolate one variable in an equation. Solve the system of linear equations using the substitution method. What were the two numbers Emma picked?

20. Aiden solved a system of equations using substitution. He stated that his solution was x = 7. Francis said something seemed wrong about the solution because it was not a point on a graph but just an x-value. Do you think Aiden’s solution is complete? If not, what else does he need to do?

21. Solve the system of equations shown below using three methods (graphing, input-output tables and substitution). y = 2x + 3 y = 15 − x

22. The Flying W Ranch raises only cows and horses. There are a total of 340 animals on the ranch. The owner prefers horses over cows so he has 52 more horses than cows. a. Write two different linear equations to model this situation. b. Solve the system of linear equations using the substitution method. How many horses live at the Flying W Ranch?

Lesson 4.4 ~ Solving Systems by Substitution 153

tic-tAc-toe ~ l e tte r to fif th gr A de rs

Write a letter to a class of fifth grade students explaining why it is important to learn math. Support your reasons with research. Give some examples of real-world situations in which they will encounter math. Include any advice you believe would help them be successful in mathematics through the middle school years. Turn in one copy of the letter to your teacher and give another copy of the letter to a fifth grade teacher in your district.

tic-tAc-toe ~ polygons

Polygons are enclosed figures whose sides are made up of line segments. Create a polygon (triangle, quadrilateral, pentagon, hexagon, etc.) by graphing linear equations that enclose the polygon. Write the equations for each line. List the vertices (or points of intersection). Color in the polygon. Repeat the process on another sheet of graph paper, creating a different polygon.

23. Tad and Timothy went to the paint store together. Tad bought 6 cans of paint and 1 paint brush for $67. Timothy bought 4 cans of the same paint and 3 of the same type of paint brushes. Timothy’s total cost was $54. a. Write a linear equation that represents Tad’s purchase and another to represent Timothy’s purchase. Let x represent the cost of a can of paint and y represent the cost of a paint brush. b. Solve the system of linear equations using the substitution method. c. What was the cost for a can of paint? The cost of a paint brush?

rEviEw

determine if the two lines in each system of equations are intersecting, parallel or the same line. state how many solutions there will be for each system.

24. y = 4 _ 5 x + 3 25. y = 1 _ 2 x + 5 26. y = 2(x + 1) y = 4 _ 5 x − 3 y = − 1 _ 2 x + 5 y = 2x + 1

27. 4x + 2y = 30 28. −x + 6y = 6 29. y = 6(x + 1) − 4 y = 4 _ 3 x − 5 y = 1 _ 6 x + 1 6x − y = 2