Before you read

Answer first — see what you already know.

Solve:
${x + y = 10 x - y = 4$

A
$(7, 3)$
B
$(6, 4)$
C
$(5, 5)$
D
$(8, 2)$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

Solve:
${3 x + 2 y = 19 3 x - 2 y = 5$
What is $y$ ?

A
$2$
B
$\frac{7}{2}$
C
$4$
D
$3$

Check your answer

Answer:

Theory

The Elimination Method

When to use: when both equations are in standard form ( $ax + by = c$ ) and the coefficients of one variable are easy to cancel.

Steps:
1. Line up the equations vertically (same variables aligned)
2. Multiply one or both equations so that one variable's coefficients are opposites
3. Add the equations to eliminate that variable
4. Solve the remaining one-variable equation
5. Back-substitute to find the other variable

$2x + 3y = 12$
$4x - 3y = 6$
$6x = 18$ add equations
$x = 3$ divide by 6

Elimination method

Example 1

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

The $y$ -coefficients are already opposites ( $+2y$ and $-2y$ ).

Add the equations:

$6x = 24 \quad \Rightarrow \quad x = 4$

Back-substitute into equation 1: $3(4) + 2y = 16$ $\to$ $2y = 4$ $\to$ $y = 2$ .

(x, y) = (4, 2)

3(4) - 2(2) = 8

✓

Example 2

Solve:
$\begin{cases} 2x + 3y = 12 \\ 4x + y = 14 \end{cases}$

Multiply equation 2 by $-3$ : $-12x - 3y = -42$

Add to equation 1:

$2x + 3y + (-12x - 3y) = 12 + (-42)$

$-10x = -30 \quad \Rightarrow \quad x = 3$

Back-substitute: $4(3) + y = 14$ $\to$ $y = 2$ .

(x, y) = (3, 2)

Theory

Choosing Which Variable to Eliminate

Look for the easiest cancellation:

1. Coefficients already opposite: just add (like $+2y$ and $-2y$ )
2. Coefficients already equal: subtract (or multiply one by $-1$ then add)
3. One coefficient is a multiple: multiply the smaller by a factor
4. Neither easy: multiply both equations by different factors

Pro tip: eliminate whichever variable requires the least multiplication.

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