1. Warm-Up Solve the system by graphing y = x + 2 x = −3 Solve the system by graphing 4x + y = 2 x − y = 3

2. Warm-Up Graph the inequality y ≤ ½x + 2 Graph the inequality y < −3

3. Section 7.2 Solving Systems by Substitution

4. Objectives Students will be able to apply the method of solving a system of linear equations by using substitution

5. Solving Systems of Equations Substitution

6. Solving by Substitution ☻S☻Step 1: Solve either equation for a variable of your choice ☻S☻Step 2: Substitute this into the other equation and solve for remaining variable. ☻S☻Step 3: Substitute this number into an equation to find the other variable.

7. Example ☻ Solve by substitution 3y + 2x = 4 -6x + y = -7 ☻ Step 1: Solve either equation for a variable of your choice -6x + y = -7 +6x y = 6x - 7 

8. Example ☻ Solve by substitution 3y + 2x = 4 -6x + y = -7 3(6x – 7) + 2x = 4  ☻ Step 2: Substitute this into the other equation and solve for the remaining variable 18x - 21 + 2x = 4 20x - 21 = 4 +21 20x = 25 20 x = 1.25 y = 6(1.25) - 7 ☻ Step 3: Substitute this number into an equation to find the other variable y = 7.5 - 7 y =.5  Answer: x = 1.25 y =.5 (1.25,.5) 3y + 2x = 4

9. Example ☻ Solve by substitution y = -4x + 8 -x + y = 7 ☻ Step 1: Solve either equation for a variable of your choice y = -4x + 8 

10. Example ☻ Solve by substitution y = -4x + 8 -x + y = 7 -x + (-4x + 8) = 7  ☻ Step 2: Substitute and solve for the other variable -5x + 8 = 7 -8 -5x = -1 -5 x =.2 y = -4(.2) + 8 ☻ Step 3: Substitute this number into the equation solved for a variable to find the remaining variable. y = -.8 + 8 y = 7.2  Answer: x =.2 y = 7.2 (.2, 7.2) -x + y = 7

11. Example Solve by substitution y = x + 3 -x + y = 4

12. Example ☻ Solve by substitution y = x + 3 -x + y = 4 ☻ Step 1: Solve either equation for a variable of your choice y = x + 3 

13. Example ☻ Solve by substitution y = x + 3 -x + y = 4 -x + (x + 3) = 4 ☻ Step 2: Substitute and solve for the remaining variable 3 = 4 3 ≠ 4 Answer: No solution Since we have eliminated our variable and we have an inequality remaining, we have a system of equations with no solutions. -x + y = 4

14. Example Solve by substitution y = ½x + 1 -x + 2y = 1 Step 1: Solve either equation for a variable of your choice

15. Example ☻ Solve by substitution y = ½x + 1 -x + 2y = 2 ☻ Step 1: Solve either equation for a variable of your choice y = ½x + 1 ☻ Step 2: Substitute this into the other equation -x + 2y = 1 -x + 2(½x + 1) = 2  

16. Example ☻ Solve by substitution y = ½x + 1 -x + 2y = 2 -x + 2(½x + 1) = 2  ☻ Step 3: Solve for the remaining variable -x + x + 2 = 2 Answer: Infinitely many solutions 2 = 2 Since we have eliminated our variable and we have a true equality remaining, we have a system of equations with Infinitely many solutions.