The linear combination method

Principle

Linear combination involves adding or subtracting the two equations (possibly after multiplying them by suitably chosen numbers) in order to eliminate one of the two unknowns.

Steps

1. Multiply one or both equations by coefficients so that the terms involving x (or y) are opposite or identical.
2. Add or subtract the two equations to eliminate one unknown.
3. Solve the resulting equation with a single unknown.
4. Substitute the value found into one of the original equations to find the second unknown.
5. Check the solution pair.

Complete example

{ 3x + 2y = 12
{ x - 2y = 4

The coefficients of y are already opposite (+2 and -2), so we can add the two equations directly:

(3x + 2y) + (x - 2y) = 12 + 4

4x = 16 -> x = 4

We substitute this back into the second equation: 4 - 2y = 4 -> -2y = 0 -> y = 0

Verification: 3(4) + 2(0) = 12 (true) and 4 - 2(0) = 4 (correct)

The solution is (4, 0).

Another example involving multiplication

{ 2x + 3y = 7
{ x + y = 3

We multiply the second equation by 2: 2x + 2y = 6

We subtract: (2x + 3y) – (2x + 2y) = 7 – 6 → y = 1

Then x = 3 – y = 2. The solution is (2, 1).

Common pitfall

When multiplying an equation by a number, you must multiply all the terms, including the one on the right-hand side (the number after the = sign). Forgetting this term is a very common mistake.