Simultaneous Equations: The Elimination Method Blueprint

By Fatima Khan, M.A. Ed·Updated April 18, 2026
Step-by-step working of simultaneous equations using the elimination method.

How do you know whether to ADD or SUBTRACT simultaneous equations?

Look at the signs of the variables you made equal. If the signs are the same (both positive or both negative), SUBTRACT the equations. If the signs are different (one positive, one negative), ADD the equations. Mnemonic: SSS (Same Signs Subtract).

Table of Contents

Simultaneous equations appear in almost every CAIE Maths paper. They are a pure test of algebraic arithmetic. While you can use substitution, the elimination method is much faster and less prone to fraction errors when dealing with linear equations. Let's break down the system from our Ultimate O-Level Mathematics Guide.

1. Scaling to Match Coefficients

You cannot add or subtract equations until the number directly in front of one of the variables (the coefficient) is identical in both equations.

EQ1: 3x + 2y = 16
EQ2: 5x - y = 18

Right now, x is (3 and 5), and y is (2 and -1). Nothing matches. But if we multiply the entire second equation by 2...

EQ1: 3x + 2y = 16
EQ3: 10x - 2y = 36 (This is EQ2 × 2)

2. Step 2: The SSS Rule (Same Signs Subtract)

Now that our 'y' coefficients are matched (2 and 2), look at their signs. One is positive (+2y) and one is negative (-2y). They have different signs.

  • SSS (Same Signs Subtract): If both were +2y, we would subtract EQ3 from EQ1.
  • DSA (Different Signs Add): Since one is +2y and the other is -2y, we ADD the two equations together.

(3x + 10x) + (2y - 2y) = (16 + 36)
13x + 0 = 52
13x = 52
x = 4

3. Step 3: Back-Substitution

Once you find the first variable (x = 4), substitute it back into EITHER of the original equations to find the second variable.

Using EQ1: 3(4) + 2y = 16
12 + 2y = 16
2y = 4
y = 2

4. Worked Exam Question (The Double Scale)

Question:

Solve the simultaneous equations:
4x + 3y = 18
3x - 5y = 28

Step 1 — Scale both equations

Neither x nor y matches easily. Let's aim to eliminate x. The lowest common multiple of 4 and 3 is 12.
Multiply EQ1 by 3: 12x + 9y = 54
Multiply EQ2 by 4: 12x - 20y = 112

Step 2 — Apply SSS Rule

Both '12x' terms are positive. Same Signs Subtract. We will subtract the bottom equation from the top.
(12x - 12x) + (9y - -20y) = (54 - 112)

Step 3 — Solve

0 + 29y = -58
y = -58 / 29
y = -2

Step 4 — Back Substitute

Using EQ1: 4x + 3(-2) = 18
4x - 6 = 18
4x = 24
x = 6

Fatima Khan📋 From the Desk of Fatima Khan
Look at Step 2 very carefully: `(9y - -20y)`. When students use the subtraction rule, they almost always mess up the double negative. They write `9y - 20y = -11y`, which ruins the rest of the problem. If you are subtracting an equation that has minus signs in it, put brackets around the second equation or physically write out `minus minus` to catch yourself!

Frequently Asked Questions

When should I use Elimination instead of Substitution?▼
Use Elimination for standard linear equations (e.g., 3x + 2y = 10). Use Substitution if you have a non-linear equation involving x², or if one equation is already arranged as x = ...
What is the SSS / DSA rule?▼
Same Signs Subtract (SSS) means if coefficients are both positive or both negative, subtract the equations. Different Signs Add (DSA) means if one is positive and one is negative, add them.
How do I check if my simultaneous equations answer is correct?▼
Substitute your x and y values back into BOTH original equations. If they balance perfectly on both sides, your answer is mathematically guaranteed to be correct.
What if the coefficients don't match?▼
Multiply one or both equations by a constant (e.g., lowest common multiple) until the numbers in front of x or y are identical.

Stop Guessing, Start Scoring

Get instant access to 500+ CAIE-aligned practice questions, worked solutions, and AI-powered mock exams across all O-Level subjects.

Access the Study Portal →Prepare with the Oracle Engine

Related Math Articles