14.1 Simultaneous equations by elimination — Learn

Prerequisites: Solve linear equations and collect like terms.

Objectives: Solve two linear equations by adding or subtracting them to remove a variable. This page covers A19.

Remove one variable

A simultaneous solution makes both equations true. Elimination combines equations so one variable disappears. Add equations when opposite coefficients cancel; subtract when matching coefficients cancel.

Example 1. Solve $2x+y=11$ and $x+y=7$.

Subtract the second equation from the first:

Substitute $x=4$ into $x+y=7$:

The solution is $(x,y)=(4,3)$. Check: $2(4)+3=11$ and $4+3=7$.

Example 2. Solve $3x+2y=16$ and $3x-y=7$.

Subtract the second equation from the first. The $x$ terms cancel:

Collect the remaining terms:

Substitute $y=3$ into $3x-y=7$ to find $x$:

Check the pair in both original equations: $3(10/3)+2(3)=16$ and $3(10/3)-3=7$.

Watch out. When subtracting an equation, subtract every term. In $(2x+y)-(x+y)$ the $y$ terms cancel because $y-y=0$. Check the final pair in both equations.

Progress check

The two taxi fares for Company A are $f+4r=14$ and $f+8r=22$. Subtract the first equation from the second. What does the result tell you about $r$?

You should now be able to

• decide whether to add or subtract equations to eliminate a variable;
• solve the remaining linear equation;
• substitute back and check both original equations.