1. The student will be able to:
Objective
The student will be able to:
solve systems of equations using elimination with multiplication.
Adapted from Skip Tyler PPT

2. Solving Systems of Equations
So far, we have solved systems using graphing, substitution, and elimination. These notes go one step further and show how to use ELIMINATION with multiplication.
What happens when the coefficients are not the same?
We multiply the equations to make them the same! You’ll see…

3. Solving a system of equations by elimination using multiplication.
Step 1: Put the equations in Standard Form.
Standard Form: Ax + By = C
Step 2: Determine which variable to eliminate.
Look for variables that have the
same coefficient.
Step 3: Multiply the equations and solve.
Solve for the variable.
Step 4: Plug back in to find the other variable.
Substitute the value of the variable
into the equation.
Step 5: Check your solution.
Substitute your ordered pair into
BOTH equations.

4. 1) Solve the system using elimination.
2x + 2y = 6
3x – y = 5
Step 1: Put the equations in Standard Form.
They already are!
No coefficients the same!
It doesn’t matter WHICH variable you change, but transform one of them so you get opposite variables
Which is easier to change? y (you only have to multiply the bottom equation by 2)
Step 2: Determine which variable to eliminate.

5. 1) Solve the system using elimination.
2x + 2y = 6
3x – y = 5
Multiply the bottom equation by 2
2x + 2y = 6
(2)(3x – y = 5)
8x = 16
x = 2
2x + 2y = 6
(+) 6x – 2y = 10
Step 3: Multiply the equations and solve.
2(2) + 2y = 6
4 + 2y = 6
2y = 2
y = 1
Step 4: Plug back in to find the other variable.

6. 1) Solve the system using elimination.
2x + 2y = 6
3x – y = 5
(2, 1)
2(2) + 2(1) = 6
3(2) - (1) = 5
Step 5: Check your solution.
Solving with multiplication adds one more step to the elimination process.

7. 2) Solve the system using elimination.
Try it before you click for answer. You ONLY get better by practicing problems.
HINT: Mult. Top by -4 and finish problem.
2) Solve the system using elimination.
x + 4y = 7
4x – 3y = 9
Step 1: Put the equations in Standard Form.
They already are!
Transform one equation so you get opposite variables.
Which is easier to change? X’s (you only have to multiply the top equation by -4 to make the x’s inverses/opposites)
Step 2: Determine which variable to eliminate.

8. 2) Solve the system using elimination.
x + 4y = 7
4x – 3y = 9
Multiply the top equation by -4
(-4)(x + 4y = 7)
4x – 3y = 9)
y = 1
-4x – 16y = -28
(+) 4x – 3y = 9
Step 3: Multiply the equations and solve.
-19y = -19
x + 4(1) = 7
x + 4 = 7
x = 3
Step 4: Plug back in to find the other variable.

9. 2) Solve the system using elimination.
x + 4y = 7
4x – 3y = 9
(3, 1)
(3) + 4(1) = 7
4(3) - 3(1) = 9
Step 5: Check your solution.

10. 3) Solve the system using elimination.
3x + 4y = -1
4x – 3y = 7
Step 1: Put the equations in Standard Form.
They already are!
Decide which variable to eliminate. Let’s use y’s because the variables are already opposite signs.
We need to multiply the top by 3 and the bottom by 4 to get a 12y and -12y.
Step 2: Determine which variable to eliminate.

11. 3) Solve the system using elimination.
3x + 4y = -1
4x – 3y = 7
Multiply both equations
(3)(3x + 4y = -1)
(4)(4x – 3y = 7)
x = 1
9x + 12y = -3
(+) 16x – 12y = 28
Step 3: Multiply the equations and solve.
25x = 25
3(1) + 4y = -1
3 + 4y = -1
4y = -4
y = -1
Step 4: Plug back in to find the other variable.

12. 3) Solve the system using elimination.
3x + 4y = -1
4x – 3y = 7
(1, -1)
3(1) + 4(-1) = -1
4(1) - 3(-1) = 7
Step 5: Check your solution.

13. What is the best number to multiply the top equation by to eliminate the x’s?
3x + y = 4
6x + 4y = 6 -4 -2 2 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

14. Solve using elimination.
Try it before you click for answer. You ONLY get better by practicing problems.
HINT: They’re already in standard form. Decide what to transform by mult.
Solve using elimination.
2x – 3y = 1
x + 2y = -3
We need to transform one equation, x’s look easier. Multiply bottom by -2.
2x – 3y = x – 3y = 1
-2(x + 2y = -3) x – 4y = 6
Now we can ADD DOWN and eliminate the x’s.
-7y =7 , y = -1
Sub back into one of the equations to find x.
x + 2(-1) = -3
x = -1 ,
Solution ( -1,-1)
2x – 3y = 1
x + 2y = -3
(2, 1)
(1, -2)
(5, 3)
(-1, -1)

15. Solve using elimination.
Try it before you click for answer. You ONLY get better by practicing problems.
HINT: Get them in standard form first, then decide what to transform by mult.
Solve using elimination.
2y = 6 - 3x get into Standard 3x + 2y = 6
x + 8 = y x – y = -8
We need to transform one equation, y’s look easier. Multiply bottom by 2.
3x + 2y = x + 2y = 6
2(x - y = -8) x – 2y = -16
Now we can ADD DOWN and eliminate the y’s.
5x = -10 , x = -2
Sub back into one of the equations to find y.
= y
y = 6 ,
Solution ( -2, 6 )
2y = 6 - 3x
x + 8 = y