Objective

1
Objective

• The student will be able to
• solve systems of equations using elimination with
  multiplication.
• SOL A.4e

Designed by Skip Tyler, Varina High School
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!
  Youll see

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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!
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.
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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.
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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.
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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.
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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.
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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.
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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.

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Which variable is easier to eliminate?
3x y 4 4x 4y 6

• x
• y
• 6
• 4

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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! Ill pick y because the signs
are already opposite.
Step 2 Determine which variable to eliminate.
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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.
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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.
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What is the best number to multiply the top
equation by to eliminate the xs?
3x y 4 6x 4y 6

• -4
• -2
• 2
• 4

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Solve using elimination.
2x 3y 1 x 2y -3

• (2, 1)
• (1, -2)
• (5, 3)
• (-1, -1)

17
Find two numbers whose sum is 18 and whose
difference 22.

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