Elimination Method

Use elimination method to solve linear systems. Math 10-C Alberta mathematics.

Lesson 5.3 of Systems of Equations in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• The Elimination Method. Learn what elimination is, how same-sign and different-sign coefficients determine whether you add or subtract equations, and the 5-step process for solving any linear system by elimination.
• Worked Examples. Walk through five fully solved examples: coefficients that already match, needing to multiply both equations, variables other than x and y, fractional answers, and systems that start with fractions that must be cleared first.

Inside the lesson — a free preview

What Is Elimination?

Definition. The elimination method means combining two equations — by adding or subtracting them — so that one variable cancels out entirely, leaving a single-variable equation you can solve directly.

For this to work, one variable must have the same coefficient or opposite coefficients in both equations. If it does not, you will need to multiply one or both equations by a constant to make that happen.

The Add or Subtract Decision

Same Signs vs. Different Signs. Once the coefficients of the variable you want to eliminate are equal in size, ask:

Same signs? → SUBTRACT the equations.

Different signs? → ADD the equations.

Memory trick: S.S.S. — Same Signs, Subtract. If the signs are different, add to cancel.

The 5-Step Process

Step 1 — Multiply to Match Coefficients. Multiply one or both equations by a constant so the coefficients of one variable are equal in absolute value.

Shortcut: sometimes the coefficients already match — in that case, skip straight to Step 2.

Step 2 — Add or Subtract the Equations. Apply the S.S.S. rule:

Same signs on the target variable → subtract the equations.

Different signs on the target variable → add the equations.

After this step, one variable disappears and you have a one-variable equation.

Step 3 — Solve the One-Variable Equation. You now have a simple equation in one unknown. Solve it using basic algebra.

Step 4 — Substitute Back to Find the Other Variable. Plug the value you just found into either of the original equations.

Solve for the remaining variable.

Step 5 — Write the Answer as an Ordered Pair. Express your solution as (x, y) (or the appropriate variable pair).

Tip: before you start, scan both equations and decide which variable will be cheaper to eliminate — pick the pair of coefficients that needs the smallest multipliers.

Worked example: Coefficients Already Match (Opposite Signs)

System to Solve. ① 2x - 5y = -1

② 3x + 5y = -14

Step 1 — Identify the Strategy. Look at the y-coefficients: -5 and +5.

They are opposite signs, so by the S.S.S. rule we ADD the equations. No multiplication needed!

Step 2 — Add the Equations. Add ① and ② column by column:

(2x + 3x) + (-5y + 5y) = (-1 + (-14))

5x + 0 = -15

5x = -15

Step 3 — Solve for x. Divide both sides by 5:

x = -3

Step 4 — Substitute into Equation ①. Replace x with -3 in 2x - 5y = -1:

2(-3) - 5y = -1

-6 - 5y = -1

-5y = -1 + 6

-5y = 5

y = -1

Answer. Solution: (-3, -1)

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