9.2 Equations with unknowns on both sides — Learn

Prerequisites: solving linear equations and collecting like terms.

Objectives: solve equations with the unknown on both sides, including brackets and decimals; recognise identities and contradictions.

This section covers equations with the unknown on both sides, including brackets, within AQA reference A17.

Collecting like terms means combining terms with the same unknown part. First expand brackets, then collect unknown terms on one side and constants on the other.

Example 1. Solve $5x+3=2x+18$.

Subtract $2x$ from both sides: $3x+3=18$.

Subtract $3$ from both sides: $3x=15$.

Divide by $3$: $x=5$.

Check: both sides equal $28$ when $x=5$.

Example 2. Solve $3(x+2)=x+14$.

Expand the bracket: $3x+6=x+14$.

Subtract $x$ from both sides: $2x+6=14$.

Subtract $6$: $2x=8$.

Divide by $2$: $x=4$.

Watch out. Do not change a term's sign without doing the same operation to both sides. From $3x+6=x+14$, subtracting $x$ and $6$ gives $2x=8$.

For example, $2x+4=2x+4$ is true for every value of $x$. In contrast, $2x+4=2x+5$ has no solution because it reduces to $4=5$.

Sometimes the unknown terms cancel. If both sides then match, every value works; if they disagree, no value works.

You should now be able to

• expand brackets and collect unknown terms;
• solve equations with unknowns on both sides;
• identify an equation with every solution or no solution.