Solve a System by Elimination with Multiplication

Problem

4x + 3y = 18,\quad 2x - y = 4

Answer

x = 3,\ y = 2

When elimination needs a multiplier, pick the variable with the simplest coefficients. Multiply one equation, make the coefficients opposites, then add.

Step-by-step solution

Choose the variable to eliminate.
$\text{eliminate } y$
Why: The y coefficients are +3 and -1, so one multiplication can line them up.
Multiply the second equation by 3.
$3(2x - y) = 3(4) \Rightarrow 6x - 3y = 12$
Why: This turns -y into -3y, the opposite of +3y.
Add the equations to cancel y.
$(4x + 3y) + (6x - 3y) = 18 + 12$
Why: The y terms add to zero.
Solve for x.
$10x = 30 \Rightarrow x = 3$
Why: Divide both sides by 10.
Substitute x = 3 into 2x - y = 4.
$6 - y = 4 \Rightarrow y = 2$
Why: The second original equation is the quickest place to find y.

Common mistakes

Multiplying only the x term by 3. Every term and the right side must be multiplied.
Adding before the coefficients are opposites, so nothing cancels.
Substituting x = 3 into the multiplied equation and then forgetting to check it in both originals.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

$3x + 2y = 16,\ x - y = 2$

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$x = 4,\ y = 2$
$2x + 3y = 13,\ x + y = 5$

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$x = 2,\ y = 3$
$5x + 2y = 19,\ x - y = 1$

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$x = 3,\ y = 2$
$3x + 4y = 18,\ x + y = 5$

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$x = 2,\ y = 3$
$4x + y = 11,\ x - 2y = 5$

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$x = 3,\ y = -1$