Solving Systems by Elimination

Solving by Elimination

By the end of this lesson, you will be able to solve a system by adding or subtracting the equations, multiply one or both first when the coefficients do not match, and choose between elimination and substitution sensibly.

Add the Equations and Watch One Vanish

Substitution replaces a variable. Elimination cancels one.

$$\begin{cases} 3x + 2y = 16 \ 5x - 2y = 8 \end{cases}$$

The $y$ terms are $+2y$ and $-2y$. Add the two equations, left to left and right to right, and they annihilate:

$$8x = 24 \longrightarrow x = 3$$

One unknown, immediately, with no rearranging and no fractions.

Why adding two equations is legal. Both sides of the second equation are the same number, so adding it to the first is adding the same thing to both sides, which has always been allowed. It is the balance rule, applied with an equation instead of a number.

Add or Subtract?

Look at the coefficients of one variable.

The pair	Do this	Because
$+2y$ and $-2y$	Add	Opposites cancel when added
$+2y$ and $+2y$	Subtract	Identicals cancel when subtracted
$-4x$ and $-4x$	Subtract	Same reasoning

Subtracting is where the errors live, because every sign on the second equation flips, including the one on the right.

$$(3x + 2y) - (x + 2y) = 14 - 6$$

$$3x + 2y - x - 2y = 8 \longrightarrow 2x = 8$$

Writing the bracket first, then removing it, is slower and correct. Doing it in your head is faster and usually wrong.

Example 1: Straight Addition

Here is the whole method on one system, worked all the way through. $$\begin{cases} 3x + 2y = 16 \ 5x - 2y = 8 \end{cases}$$

• Which variable will cancel if you add the equations? y
• Add them. What equation is left? 8x = 24
• Solve for x 3
• Back-substitute into the first equation to find y 3.5
• Check in the SECOND equation. What does 5x - 2y come to? 8

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