1. Topic 6.5. Solve Systems by Substitution Objectives: Solve Systems of Equations using Substitution Standards: Functions, Algebra, Patterns. Connections and Representations.

2. Solving Systems by Substitution 4 There are five steps to solving a system by substitution. 4 1) Solve one equation for one variable (y = ; x = ; a =) 4 2) Substitute the value for the variable in the other equation. 4 3) Simplify and solve the equation. 4 4) Substitute back into either original equation to find the value of the other variable. 5) Check the solution in both equations of the system.

3. Example # 1: Solve x = 1 - 4y, 2x + 7y = 3 using substitution. 4 x = 1 - 4y 4 2x + 7y = 3 4 1) x = 1 - 4y 4 2) 2x + 7y = 3 2(1 - 4y) + 7y = 3 2 - 8y + 7y = 3 4 3) 2 - 8y + 7y = 3 2 - y = 3 - 2 - 2 - y = 1 y = - 1 4 1) The first equation is already solved for x, so choose it for step 2. 4 2) Substitute 1 - 4y for x in the other equation. You can distribute and then combine like terms. 4 3) Solve for the other variable 4 y = - 1.

4. Solve x = 1 - 4y, 2x + 7y = 3 using substitution. 4 4) x = 1 - 4y x = 1 - 4(- 1) x = 1 + 4 x = 5 The solution is (5,- 1). 4 4) Now use this value in either equation to find the other variable.

5. 5 ) Check the solution (5, -1), by substituting in both equations. 4 x = 1 - 4y 5 = 1 - 4( - 1) 5 = 1 + 4 5 = 5 Since (5, -1) works in both equations it is the solution. 4 2x + 7y = 3 2(5) + 7( -1) = 3 10 - 7 = 3 3 = 3

6. Example #2: y = 4x 3x + y = -21 Step 1: Solve one equation for one variable. y = 4x (This equation is already solved for y.) Step 2: Substitute the expression from step one into the other equation. 3x + y = -21 3x + 4x = -21 Step 3: Simplify and solve the equation. 7x = -21 x = -3

7. y = 4x 3x + y = -21 Step 4: Substitute back into either original equation to find the value of the other variable. 3x + y = -21 3(-3) + y = -21 -9 + y = -21 y = -12 Solution to the system is (-3, -12).

8. y = 4x 3x + y = -21 Step 5: Check the solution in both equations. y = 4x -12 = 4(-3) -12 = -12 3x + y = -21 3(-3) + (-12) = -21 -9 + (-12) = -21 -21= -21 Solution to the system is (-3,-12).

9. Example #2: x + y = 10 5x – y = 2 Step 1: Solve one equation for one variable. x + y = 10 y = -x +10 Step 2: Substitute the expression from step one into the other equation. 5x - y = 2 5x -(-x +10) = 2

10. x + y = 10 5x – y = 2 5x -(-x + 10) = 2 5x + x -10 = 2 6x -10 = 2 6x = 12 x = 2 Step 3: Simplify and solve the equation.

11. x + y = 10 5x – y = 2 Step 4: Substitute back into either original equation to find the value of the other variable. x + y = 10 2 + y = 10 y = 8 Solution to the system is (2,8).

12. x + y = 10 5x – y = 2 Step 5: Check the solution in both equations. x + y =10 2 + 8 =10 10 =10 5x – y = 2 5(2) - (8) = 2 10 – 8 = 2 2 = 2 Solution to the system is (2, 8).

13. Now you try! Solve by substitution. 1. 2. There are five steps to solving a system by substitution. 1) Solve one equation for one variable (y= ; x= ; a=) 2) Substitute the value for the variable in the other equation. 3) Simplify and solve the equation. 4) Substitute back into either original equation to find the value of the other variable. 5) Check the solution in both equations of the system.