Systems of Equations: Elimination Method

What is the elimination method?

The elimination method (also called the "addition method") works by adding or subtracting the two equations so that one variable cancels out. The key insight: if one variable has the same (or opposite) coefficient in both equations, adding or subtracting eliminates it.

Step-by-step solution

We have:

• Equation 1: $2x + 3y = 12$
• Equation 2: $4x - 3y = 6$

Step 1 — Check if any variable has matching/opposite coefficients. Look at $y$: Equation 1 has $+3y$ and Equation 2 has $-3y$. These are opposites — adding the equations will eliminate $y$.

Step 2 — Add the equations.

$\begin{aligned} 2x + 3y &= 12 \\ + \quad 4x - 3y &= 6 \\ \hline 6x + 0y &= 18 \end{aligned}$

The $y$ terms cancel: $+3y + (-3y) = 0$.

Step 3 — Solve for $x$.

$6x = 18 \implies x = 3$

Step 4 — Back-substitute into either original equation. Using Equation 1:

$2(3) + 3y = 12 \implies 6 + 3y = 12 \implies 3y = 6 \implies y = 2$

Step 5 — Solution: $(x, y) = (3, 2)$.

Step 6 — Check in Equation 2: $4(3) - 3(2) = 12 - 6 = 6$ ✓

What if coefficients don't match?

Multiply one or both equations by constants to create matching/opposite coefficients. For example, $x + 2y = 5$ and $3x + y = 10$: multiply the first equation by $-3$ to get $-3x - 6y = -15$, then add to the second: $-5y = -5$, $y = 1$.

Common mistakes

• Adding when you should subtract (or vice versa). If both coefficients are the same sign (e.g., $+3y$ and $+3y$), you need to subtract (or multiply one equation by $-1$ first). If they're opposite signs, add.
• Forgetting to multiply ALL terms. When multiplying an equation by a constant, every term (including the constant on the right) must be multiplied.

Try it in the visualization

Adjust the six coefficients ($a_1, b_1, c_1, a_2, b_2, c_2$). Watch the two equations line up for elimination. The animation shows the equations being added, the cancelled variable disappearing, and the resulting single equation being solved.