5C Solving by Elimination
1) Determine a number that the ‘x’ coefficient of both equations go into
2) Multiply each equation by a number that will make the ‘x’ coefficient. Be sure to make one
equation positive and one equation negative
3) Add both equations
4) Solve the simple equation for y
5) Substitute the ‘y’ into one equation and solve for x.
6) Put answer in (x,y)
Example 1
Example 2
(A) 2x + 3y = 14
(A) 5x – y = 17
(B) 4x - 2y = -4
(B) 3x + 2y = 5
2x and 4x go into 4x so multiply (A)
5x and 3x go into 15x so multiply (A) by 3 and (B) by -5
by 2 and (B) by -1
(A) 2(2x + 3y = 14)  4x + 6y = 28
(B) -1(4x – 2y = -4)  -4x + 2y = 4
8y = 32
y=4
(A) 2x + 3(4) = 14
2x + 12 = 14
2x = 2
x = 1 (1,4)
(A) 3(5x – y = 17)  15x – 3y = 51
(B) -5(3x + 2y = 5)  -15x – 10y = -25
-13y = 26
y = -2
(A) 5x – (-2) = 17
5x + 2 = 17
5x = 15
x = 3 (3, -2)
When using elimination: If both variables disappear then the answer is either no solution (0x + 0y = #)
Or infinitely many (0x + 0y = 0)
Example 3:
Example 4:
Example 5
2x + 3y = 12
6x + 2y = 12
8x + 3y = -16
-2x - 3y = -14
-6x – 2y = -12
-8x + 5y = 16
0x + 0y = -2 so no solution
0x + 0y = 0 so infinitely many
8y = 0 so y = 0.
Problems
0: 2x + 3y = 3
5(2x + 3y = 3)
10x + _____ = 15
2x + 3( ) = 3
5x – y = 16
-2(5x – y = 16)
-10x + _____ = __
2x + _____= 3
2x = 6
_____ = ____ so y = -1
x = ___
1. Solve for x if y = -3 for 5x + 2y = 9
2. Solve for x if y = -4 for -2x – y = 10
3-4. What number do the ‘x’ coordinates go into 5.What must you multiply the top by to eliminate the x?
3. 5x – 2y = 10
4. -7x + 2y = 10
5. 2x – 5y = 12
10x – 7y = 12
3x + 4y = 19
-6x + y = 18
6- 17 Solve the following system of equations:
6.
2x + 3y = 6
7. 2m + 3n = 4
x + 2y = 5
-m + 2n = 5
10. 4x – 2y = 10
x+y =4
14. 3x – 7y = 12
3y = 8 – 2x
8. 3a – b = 2
a + 2b = 3
9. 3x – 5y = 12
12x – 20y = 48
11. 10x – 20y = 0
5x – 10y = -16
12. 3x – 2y = 7
4x + 3y = 32
13. x + y = 12
x–y=4
15. 2(x + 3y) = 14
x – y = -5
16. 2x = 3y + 1
4x = 3y + 5
17. 8x – 2y = 1
-4x + 6y = 7
18. Solve for x in the following equation
5x + 3y = 22
-3x + y = -2
19. Solve for y in the following equation:
9x + 8y = 33
3x – 2y = -3
20. The price of 4 sodas and 5 snacks is $15.50. The cost of 3 sodas and 2 snacks is $9. Find the cost of each.
21. Three medium pizzas and two large pizzas is $58. Two medium pizzas and one large pizza is $34. What
is the cost of an order that has seven medium pizzas and three large pizzas?