9.3 Forming and solving equations — Learn

Prerequisites: forming expressions and solving equations from Sections 9.1 and 9.2.

Objectives: define an unknown, form an equation from words, solve it, and interpret the answer in context.

This section covers forming and solving equations from simple situations, within AQA reference A21.

Choose a letter for the unknown and state what it represents. Translate each relationship into an expression. Use the equality in the question to form an equation, then check that your answer fits the situation.

Example 1. A number and the next integer total $31$. Find the numbers.

Let the smaller integer be $n$; the next is $n+1$.

Form the equation: $n+(n+1)=31$.

Collect terms: $2n+1=31$.

Subtract $1$: $2n=30$.

Divide by $2$: $n=15$.

The integers are $15$ and $16$; their total is $31$.

Example 2. A rectangle has length $3$ cm more than its width. Its perimeter is $34$ cm.

Let the width be $w$ cm; the length is $(w+3)$ cm.

Perimeter is twice the length plus twice the width: $2w+2(w+3)=34$.

Expand and collect: $4w+6=34$.

Subtract $6$: $4w=28$.

Divide by $4$: $w=7$.

The dimensions are $7$ cm and $10$ cm. Their perimeter is $34$ cm.

Watch out. “Three more than a number” is $n+3$, not $3n$. Test the expression with $n=5$: three more is $8$, while three times is $15$.

Progress check

For the hall prices, let $h$ be the number of hours. Set $18+6h=10+8h$. Solve to find the equal-price time, then compare both costs.

You should now be able to

• define an unknown and state what it represents;
• translate a relationship into an equation;
• solve and check the result in its context.