Solving Linear Systems by Substitution. ESSENTIAL QUESTION How can you solve a system of linear equations by using substitution? A-REI.3.

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L E S S O N 9.2 Solving Linear Systems by Substitution A-REI.3.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

1 L E S S O N 9.2 Solving Linear Systems by Substitution A-REI.3.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Also A-CED.1.3? ESSENTIAL QUESTION How can you solve a system of linear equations by using substitution? EXPLORE ACTIVITY A-REI.3.6 Solve by Substituting In the system of linear equations shown below, the value of y is given. You can use this value of y to find the value of x and the solution of the system. A { y = 3 x + y = 5 Substitute the value for y in the second equation and solve for x. x + y = 5 x + = 5 y O x x = B You know the values of x and y. What is the solution of the system? (, ) C Graph the system of linear equations. How do your solutions compare? D Use substitution to find the values of x and y in this system of linear equations. Once you find the value for x, substitute it either original equation to find the value for y. y = 2x { 2x + 3y = 40 REFLECT (, ) 1. For the system in part D, what equation did you get after substituting 2x for y in 2x + 3y = 40 and simplifying? 2. How could you check your solution in part D? Lesson

Math On the Spot Solve a Linear System by Substitution The substitution method is used to solve systems of linear equations by solving an equation for one variable and then substituting the resulting

Substitute the expression from Step 1 into the other equation and solve for the other variable. 3.

2 Math On the Spot Solve a Linear System by Substitution The substitution method is used to solve systems of linear equations by solving an equation for one variable and then substituting the resulting expression for that variable into the other equation. The steps for this method are as follows: 1. Solve one of the equations for one of its variables. 2. Substitute the expression from Step 1 into the other equation and solve for the other variable. 3. Substitute the value from Step 2 into either original equation and solve to find the value of the variable in Step 1. EXAMPLE 1 A-REI.3.6 My Notes Solve the system of linear equations by substitution. Check your answer. { -3x + y = 1 4x + y = 8 STEP 1 Solve an equation for one variable. 3x + y = 1 Select one of the equations. y = 3x + 1 Solve for the variable y. Isolate y on one side. STEP 2 Substitute the expression for y in the other equation and solve. 4x + 3x + 1 = 8 Substitute the expression for the variable y. 7x + 1 = 8 Combine like terms. 7x = 7 Subtract 1 from each side. x = 1 Divide each side by 7. STEP 3 Substitute the value of x you found into one of the equations and solve for the other variable, y. 3(1) + y = 1 Substitute the value of x into the first equation. 3 + y = 1 Simplify. y = 4 Add 3 to each side. So, (1, 4) is the solution of the system. STEP 4 Check the solution by graphing. 3x + y = 1 4x + y = 8 x-intercept: - 1_ 3 x-intercept: 2 y-intercept: 1 y-intercept: 8 The point of intersection is (1, 4) x + y = 1-4 O 2 y 4x + y = 8 4 x 286 Unit 2A

{ x + 3y = 4 -x + 2y = 6 Personal Math Trainer Online Assessment and Intervention Solving Special Systems by Substitution You can use the substitution method for systems of linear equations that have

3 REFLECT 3. Justify Reasoning Is it more efficient to solve 3x + y = 1 for x? Why or why not? 4. What is another way to check your solution? YOUR TURN Solve each system of linear equations by substitution. Check your answer. 5. { 2x + y = 5 y = x { x + 3y = 4 -x + 2y = 6 Personal Math Trainer Online Assessment and Intervention Solving Special Systems by Substitution You can use the substitution method for systems of linear equations that have infinitely many solutions and for systems that have no solutions. EXAMPLE 2 Solve each system of linear equations by substitution. A-REI.3.6 Math On the Spot A { x - y = -x + y = 4 STEP 1 Solve x y = 2 for x: x = y 2 STEP 2 Substitute the resulting expression into the other equation and solve. (y 2) + y = 4 Substitute. 2 4 Simplify. The resulting equation is false, so the system has no solutions y -x + y = 4 STEP 3 Graph the equations to provide more information. The graph shows that the lines are parallel and do not intersect. -4 x - y = 2 O 2 4 x Lesson

4 My Notes B { 2x + y = 4x + 2y = -4 STEP 1 Solve 2x + y = for y: y = x - 2 STEP 2 Substitute the resulting expression into the other equation and solve. y 4x + 2(x - 2) = -4 Substitute. 2 Use the Distributive 4x - 4x - 4 = -4 Property. O = -4 Simplify. x STEP 3 The resulting equation is true so the system has infinitely many solutions. Graph the equations to provide more information The graphs are the same line, so the system has infinitely many solutions. REFLECT 7. In part B of Example 2, why is it more efficient to solve and substitute for y than to solve and substitute for x? 8. Give two possible solutions of the system in part B of Example 2. How are all the solutions of this system related to one another? Personal Math Trainer Online Assessment and Intervention YOUR TURN Solve each system of linear equations by substitution. 9. { x + 3y = 6 2x + 6y = 12 2x - y = { 2x - y = Unit 2A

Modeling with Linear Systems You can use a system of linear equations and its graph to model many real-world situations. EXAMPLE 3 A-REI.3.6, A-CED.1.

In how many months will both fitness centers cost the same? What will that cost be? Write an equation for each option.

Option 1 t = 50 + 30 m Option 2 t = 0 + 40 m STEP 1 t = 50 + 30m t = 40m Write the system of equations.

STEP 3 t = 40m = 40(5) = 200 STEP 4 (5, 200) In 5 months, the total cost for each option will be the same, $200. YOUR TURN Write one of the original equations. Substitute 5 for m.

5 Modeling with Linear Systems You can use a system of linear equations and its graph to model many real-world situations. EXAMPLE 3 A-REI.3.6, A-CED.1.3 One family fitness center has a $50 enrollment fee and costs $30 per month. Another center has no enrollment fee and costs $40 per month. In how many months will both fitness centers cost the same? What will that cost be? Write an equation for each option. Let t represent the total amount paid and m represent the number of months. Math On the Spot Total cost is enrollment fee plus cost per month times months. Option 1 t = m Option 2 t = m STEP 1 t = m t = 40m Write the system of equations. STEP m = 40m -30m -30m 50 = 10m 50 = 10m = m Substitute m for t in the second equation. Subtract 30m from each side. Divide each side by 10. STEP 3 t = 40m = 40(5) = 200 STEP 4 (5, 200) In 5 months, the total cost for each option will be the same, $200. YOUR TURN Write one of the original equations. Substitute 5 for m. Write the solution as an ordered pair. 11. One high-speed Internet provider has a $30 setup fee and charges $40 per month. Another provider has a $60 setup fee and charges $30 per month. In how many months will the cost be the same? Math Talk Mathematical Practices In a graph of the system of equations, why must the values of the variables be greater than or equal to zero? Personal Math Trainer Online Assessment and Intervention Lesson

6 Guided Practice Solve each system of linear equations by substitution. Check your answer. (Examples 1 and 2) y = x { 2. 4x + y = 20 { x + 2y = 6 2x + 4y = 12 STEP 1 Find the value of x. STEP 1 Solve x + 2y = 6 for x. 4x + y = 20 x = 4x + = 20 STEP 2 Substitute that expression for x into 2x + 4y = 12. x = 2 ( ) + 4y = 12 x = - y + y = 12 STEP 2 Find the value of y. y = x + 5 y = + 5 = 12 y = Solve each system of linear equations by substitution. Check your answer. (Examples 1 and 2) 3. { x + 2y = 7 2x - y = -4 4x + 3y = 3 4. { 2x - 2y = { y = 2x y = 3x - 3? 6. The Blanco family is deciding between two lawn-care services. Evergreen charges a $49 startup fee, plus $29 per month. Great Grass charges a $25 startup fee, plus $37 per month. In how many months will both lawn-care services cost the same? What will that cost be? (Example 3) ESSENTIAL QUESTION CHECK-IN 7. Explain how you can solve a system of linear equations by substitution. 290 Unit 2A

7 Name Class Date 9.2 Independent Practice A-REI.3.6, A-CED.1.3 Personal Math Trainer Online Assessment and Intervention For each linear system, tell whether it is more efficient to solve for x and then substitute for x or to solve for y and then substitute for y. Explain your reasoning. Then solve the system. 8. 6x - 3y = 15 { x + 3y = Communicate Mathematical Ideas The solution of a system of two linear equations yields the equation 0 = 3. Describe what the graph of the system looks like. 9. 3x - y = -1 { 5x - y = Use one solution, no solutions, or infinitely many solutions to complete this statement. When the solution of a system of linear equations yields the equation 4 = 6, the system has. 14. Represent Real-World Problems Ella buys a book and a pen for $14. The cost of the book is $2 more than twice the cost of the pen. Write a system of linear equations for the situation. Then find the cost of each item. Let x represent the cost of the pen, and let y represent the cost of the book. For each system of linear equations, write the expression you could substitute for x. Then solve the system. 10. { 2x - y = 6 x + y = { x - 2y = 0 4x - 3y = Interpret the Answer The perimeter of a rectangular picture frame is 66 inches. The length is 3 inches greater than the width. The system of linear equations used to represent the situation is shown below. 2x + 2y = 66 { x + 3 = y Name the dimension represented by each variable. Lesson

16. Kim and Leon exercise a total of 20 hours each week. Leon exercises 2 hours less than 3 times the number of hours Kim exercises. How many hours does each exercise? 17.

CINEMA SNAKSHAK Customer #3598 3 large popcorn buckets 2 small drinks total due: $21.00 CINEMA SNAKSHAK Customer #3599 2 large popcorn buckets 4 small drinks total due: $22.

8 16. Kim and Leon exercise a total of 20 hours each week. Leon exercises 2 hours less than 3 times the number of hours Kim exercises. How many hours does each exercise? 17. Multi-Step Use the receipts below to write and solve a system of linear equations to find the cost of a large popcorn bucket and the cost of a small drink. CINEMA SNAKSHAK Customer # large popcorn buckets 2 small drinks total due: $21.00 CINEMA SNAKSHAK Customer # large popcorn buckets 4 small drinks total due: $22.00 FOCUS ON HIGHER ORDER THINKING Work Area 18. Multiple Representations For the first equation in the system of linear equations below, write an equivalent equation without denominators. Then solve the system. { 2 x + y 3 = 6 x - y = Critical Thinking Is it possible for a system of three linear equations to have one solution? If so, give an example. 20. Draw Conclusions Solve the system of linear equations below. What does the solution tell you about the graph of the system? x - 2y = -4 { 4y = 2x + 8 Image Credits: Artville/Getty Images, Fotofermer/Shutterstock 292 Unit 2A

Solving Systems by Substitution

6-2 Solving Systems by Substitution Objective Solve systems of linear equations in two variables by substitution. Why learn this? You can solve systems of equations to help select the best value among