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Objective

The student will be able to:

solve systems of equations using elimination with multiplication.

Chapter 4.2

Designed by Skip Tyler, Varina High School

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…

Solving a system of equations by elimination using multiplication.

Standard Form: Ax + By = C

Step 1: Put the equations in Standard Form.

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.

1) Solve the system using elimination.

2x + 2y = 6

3x – y = 5

Step 1: Put the equations in Standard Form.

They already are!

None of the coefficients are the same!

Find the least common multiple of each variable.

LCM = 6x, LCM = 2y

Which is easier to obtain?

2y(you only have to multiplythe bottom equation by 2)

Step 2: Determine which variable to eliminate.

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.

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.

2) Solve the system using elimination.

x + 4y = 7

4x – 3y = 9

Step 1: Put the equations in Standard Form.

They already are!

Find the least common multiple of each variable.

LCM = 4x, LCM = 12y

Which is easier to obtain?

4x(you only have to multiplythe top equation by -4 to make them inverses)

Step 2: Determine which variable to eliminate.

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.

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.

What is the first step when solving with elimination?

- Add or subtract the equations.
- Multiply the equations.
- Plug numbers into the equation.
- Solve for a variable.
- Check your answer.
- Determine which variable to eliminate.
- Put the equations in standard form.

3) Solve the system using elimination.

3x + 4y = -1

4x – 3y = 7

Step 1: Put the equations in Standard Form.

They already are!

Find the least common multiple of each variable.

LCM = 12x, LCM = 12y

Which is easier to obtain?

Either! I’ll pick y because the signs are already opposite.

Step 2: Determine which variable to eliminate.

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.

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.

What is the best number to multiply the top equation by to eliminate the x’s?

- -4
- -2
- 2
- 4

3x + y = 4

6x + 4y = 6

Find two numbers whose sum is 18 and whose difference 22.

- 14 and 4
- 20 and -2
- 24 and -6
- 30 and 8

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