Procedure

Eliminate one variable from a linear pair

Algebra • Equations • Simultaneous Equations

Statement

Given

a_{1} x + b_{1} y = c_{1}

and

a_{2} x + b_{2} y = c_{2}

, multiply each equation by a constant chosen to make the coefficients of one variable equal, then subtract the equations to remove that variable; the result is a single linear equation in the other variable, and back-substitution gives the first. The method needs

a_{1} b_{2} - a_{2} b_{1} \neq = 0

, which is exactly the condition for a unique solution.

How to use it

1Look first for a variable whose coefficients are already equal or opposite — then no scaling is needed and one subtraction finishes it.
2When the two equations share a constant term or a total, subtracting them directly often removes a variable without any scaling at all.
3Substitution is the same method in disguise: use it when one equation already has a variable with coefficient $1$ .
4Always back-substitute into the equation you did not use last, so the check is independent of the step that produced the value.

Worked example

Solve

2 x + 3 y = 13

and

x + y = 5

Step-by-step

1
$2 (x + y) = 2 (5) \Rightarrow 2 x + 2 y = 10$
Why: Scale the second equation so that the coefficient of $x$ matches the $2 x$ in the first.
2
$(2 x + 3 y) - (2 x + 2 y) = 13 - 10 \Rightarrow y = 3$
Why: Subtracting removes $x$ and leaves one unknown.
3
$x + 3 = 5 \Rightarrow x = 2$
Why: Back-substitute into the simpler of the two equations.

Answer

x = 2, y = 3

Watch out

If the scaling makes both variables cancel, the pair has either no solution or infinitely many;

a_{1} b_{2} = a_{2} b_{1}

is the warning sign, and the consistency condition decides which. Subtracting equations is where sign slips live: subtract the whole right-hand side too, not just the left.

Builds on

Turn a worded condition into a linear equation in one unknown

Also known as

elimination method, solve simultaneous linear equations, add and subtract equations, substitution method, two equations two unknowns

Eliminate one variable from a linear pair

Statement

Worked example

Appears in past papers

2020 (II)

2020 (I)