Substitution method

In the substitution method we make one unknown the subject of one equation, then substitute that expression into the other equation. This leaves one equation in one unknown.

Why it is allowed: for the solution pair, the unknown really is equal to that expression, so it can be replaced by the expression wherever it appears in the other equation.

Method

1. Choose an equation in which an unknown has coefficient \(1\) or \(-1\), and make that unknown the subject.
2. Substitute the expression, in brackets, into the other equation.
3. Expand, simplify and solve for the remaining unknown.
4. Substitute this value into the expression from step 1 to find the other unknown.
5. Check the pair in both original equations.

Example

Solve \(2x+y=9\) \((1)\) and \(7x-5y=23\) \((2)\).

From \((1)\): \(y=9-2x\) \((3)\)

\(\begin{aligned} 7x-5(9-2x)&=23 \\ 7x-45+10x&=23 \\ 17x&=68 \\ x&=4 \end{aligned}\)

From \((3)\): \(y=9-2(4)=1\). Check in \((2)\): \(7(4)-5(1)=28-5=23\). The solution is \(x=4\), \(y=1\).

Check: substitute into the other equation, not the one you rearranged. Substituting \(y=9-2x\) back into \((1)\) gives \(2x+9-2x=9\), so \(9=9\), which is true but tells you nothing. Keep the brackets: \(-5(9-2x)=-45+10x\), not \(-45-10x\).