1. 8.3 The Addition Method
Also referred to as Elimination Method

2. The Addition Method Write the second equation below the first.
Add the equations together and solve for the remaining variable.

3. Solve: + x + y = 5 x - y = 1 x - y = 1 x - y = 1 x - y = 1 x - y = 1
= 6
x = 3
3 + y = 5
y = 2
(3,2)

4. x + y = 5
x + y = 5
2x - y = 4 3x
= 9
x = 3
3 + y = 5
y = 2
(3,2)

5. 2 Special Cases No solution Case 1: NO SOLUTION
Both variables cancel out but the constant does not
Leaving a false equation.
4x - 2y = 2
-4x + 2y = -16
0 + 0 = -14
No solution

6. Case 2: INFINITELY MANY SOLUTIONS
Both variables and the constant cancel out
Leaving a true equation
5x - 7y = 6
-5x + 7y = -6
0 + 0 = 0
Infinitely many solutions

7. They all do not cancel out so easily
If the equations do not eliminate a variable when you add them together:
Multiply the whole equation by a number that will help you cancel it out.

8. 2 4x - 2y = 7 4x - 2y = 7 3x + y = 4 6x + 2y = 8 10x = 15 x = 1.5
(1.5,-0.5)

9. -1 2x + 3y = 8 2x + 3y = 8 x + 3y = 7 -x - 3y = -7 x = 1 2(1) + 3y = 8
(1,2)

10. 3 4 4x + 2y = 18 12x + 6y = 54 -3x + 5y = 6 -12x + 20y = 24 26y = 78
(3,3)
y = 3

11. 3 -5 5x + 3y = 2 15x + 9y = 6 -15x - 35y = 20 3x + 7y = -4 -26y = 26
(1,-1)
x = 1

12. 5(a-b)=10 and a+b=2

13. The sum of two numbers is 72. The difference is 58. Find the numbers.
x + y = 72
x - y = 58
7 and 65

14. The sum of the length and width of a rectangle is 25 cm
The sum of the length and width of a rectangle is 25 cm. The length is 2 less than twice the width. Find the length and width.

15. Assignment: Page 371 (2-40) even