Elimination by adding or subtracting

In the elimination method we add or subtract the two equations so that one unknown disappears. This works at once when that unknown has coefficients of the same size in both equations.

Why it is allowed: the solution pair makes both equations true, so the two left-hand sides are equal to the two right-hand sides. Adding equal amounts to equal amounts gives equal totals, and the same holds for subtracting, so the new equation is true for the same pair. Choosing to add or subtract so that one unknown cancels leaves an equation in one unknown.

Method

1. Write the equations one under the other and label them \((1)\) and \((2)\).
2. Choose the unknown whose coefficients are the same size. Subtract the equations if the signs are the same; add them if the signs are opposite.
3. Solve the resulting equation for the remaining unknown.
4. Substitute this value into either original equation to find the other unknown.
5. Check the pair in both original equations.

Example

Solve \(7x+3y=29\) \((1)\) and \(5x-3y=-5\) \((2)\).

The \(y\) terms are \(+3y\) and \(-3y\): the same size with opposite signs, so add.

\(\begin{aligned} (1)+(2):\quad 12x&=24 \\ x&=2 \end{aligned}\)

Substitute \(x=2\) into \((1)\): \(14+3y=29\), so \(3y=15\) and \(y=5\).

Check in \((2)\): \(5(2)-3(5)=10-15=-5\). The solution is \(x=2\), \(y=5\).

Check: when you subtract, subtract every term of the second equation, including the constant. The commonest error is with negative terms: \(3y-(-2y)=5y\), not \(y\).