14.2 Simultaneous equations by substitution — Learn

Prerequisites: Solve linear equations and substitute values into expressions.

Objectives: Solve pairs of linear equations by replacing a variable with an equal expression. This page covers A19.

Replace a variable

Substitution means replacing a variable with an expression equal to it. Use this method when one equation already gives a variable on its own.

Example 1. Solve $y=2x-1$ and $x+y=8$.

Replace $y$ in the second equation with $2x-1$:

Substitute $x=3$ into $y=2x-1$:

The solution is $(3,5)$. Check: $3+5=8$.

Example 2. Solve $x=3y+2$ and $2x+y=19$.

Replace $x$ with $3y+2$ in the second equation:

Then $x=3(15/7)+2=59/7$. Check by substitution in both original equations.

Watch out. If $y=2x-1$, substitute the whole expression using brackets: $x+(2x-1)$, not $x+2x-1$ without tracking the signs. Expand and check the result in both equations.

Progress check

For Company B, the fares give $f+4r=16$ and $f+8r=24$. Use the first equation to write $f$ in terms of $r$. What equation do you get after substitution?

You should now be able to

• rearrange a linear equation to make one variable the subject;
• substitute an expression into the other equation;
• find both values and check both equations.