1. 7.2, 7.3 Solving by Substitution and Elimination

2. Solving by Substitution
1) Use one equation to get either x or y by itself
Substitute what we just found in step 1 to the other equation
Solve for one unknown
Substitute the answer from step 3 to either of the original equation to solve for the other unknown
Check your answer

3. x + y = 4
x = y + 2
Step 1) x is already by itself x = y + 2
Step 2) Substitute to the first equation
x + y = 4
(y + 2) + y = 4
Step 3) y + 2 = 4
2y = 4 – 2 = 2
y = 1
Step 4) x = y + 2
x = = 3
Solution: (3, 1)

4. 2x + y = 6
3x + 4y = 4
Step 1) get y by itself 2x + y = 6 so y = -2x + 6
Step 2) Substitute to the second equation
3x + 4y = 4
3x + 4(-2x + 6) = 4
Step 3) 3x – 8x = 4
-5x = 4
-5x = -20
x = 4
Step 4) 2x + y = 6
2(4)+y = 6 so 8 + y = 6, y = -2
Solution: (4, -2)

5. More practice: (Make sure you check your answers)
1) 3x – 4y = 14
5x + y = 8
Answer: (2,-2)
2) 2x – 3y = 0
-4x + 3y = -1
Get x by itself: 2x – 3y = 0 so x = (3/2) y
Substitute into: -4x y = -1
-4(3/2)y + 3y = -1
-6y y = -1
- 3y = -1 so y = 1/3
Solve for x: Use the first equation 2x – 3y = 0
2x – 3(1/3) = 0
2x – = 0
x = ½
Answer: (1/2 , 1/3)

6. Example 1
The perimeter of a basketball court is 288 ft. The length is 40 ft longer than the width. Find the dimensions of the court.

7. Example 2
The sum of two numbers is 51. One number is 27 more than the other. Find the numbers.

8. Solve by Elimination
Multiply some numbers to either or both equations to get 2 opposite terms (For example: 2x and -2x)
Add equations to eliminate one variable
Solve for 1 unknown
Substitute the answer from step 3 to either of the original equation to solve for the other unknown
Check your answer

9. Solve by elimination 2x – 3y = 0 -4x + 3y = -1
Ignore step 1 since we already have 2 opposite terms -3y and 3y
Step 2 &3: Add 2 equations to eliminate y and solve for x
-2x = -1
x = 1/2
Step 4: Choose the first equation 2x – 3y = 0
2(1/2) – 3y = 0
y = 0
- 3y = -1
y = 1/3
Answer: ( 1/2, 1/3)

10. Ex2) 2x + 2y = 2
3x – y = 1
Step 1: Multiply 2 to the second equation
2x + 2y = 2
(2) 3x – y = 1
6x – 2y = 2
8x = (step 2 and 3)
x = 1/2
Step 4: Choose 3x – y = 1
3(1/2) – y = 1
3/2 - y = 1
- y = 1 – (3/2) = - ½
y = ½
Answer: (1/2, 1/2)

11. Ex3) 2x + 3y = 8
-3x + 2y = 1
Step 1: Multiply 3 to the 1st equation and 2 to the 2nd equation
(3) 2x + 3y = 8
(2) -3x + 2y = 1
6x + 9y = 24
-6x + 4y = 2
13y = (step 2 and 3)
y = 2
Step 4: Choose -3x + 2y = 1
-3x + 2(2) = 1
-3x = 1
-3x = = -3
x = 1
Answer: (1,2)

12. Practice Solve by elimination 1) 3x – 4y = 14 5x + y = 8
Answer: (2,-2)
3x + 2y = 7
6x + 4y = 14
Answer: Infinite many solutions