Substitution and Formulae

Substitution means replacing a letter with a number, then working out the answer. At Stage 8 the big new idea is substituting negative numbers, and the safest habit is to wrap each value in brackets as you put it in.

Take the expression $2a + b$ with $a = 5$ and $b = -3$. Substitute carefully: $2a + b = 2 \times (5) + (-3) = 10 - 3 = 7.$

The method is always the same:

1. Write the expression.
2. Replace each letter with its value, using brackets.
3. Work out the answer, following BIDMAS.

Remember the hidden multiplication sign: $3n$ means $3 \times n$. So if $n = -4$, then $3n = 3 \times (-4) = -12$. Putting the times sign back in keeps you safe.

Negative substitution leans on the rules you already know for directed numbers. Once the value is in the expression, the calculation follows those rules.

The two rules that matter most:

• A negative times a negative is positive: $(-2) \times (-3) = 6$.
• A negative times a positive is negative: $4 \times (-3) = -12$.

Two worked snippets with $n = -5$:

• $2n = 2 \times (-5) = -10$.
• $5 - n = 5 - (-5) = 5 + 5 = 10$ — subtracting a negative adds.

Take it slowly: write the bracketed value first, then apply the sign rule, then finish the arithmetic.

Powers and brackets need the most care of all, especially with negatives.

A key fact: a negative number squared is positive, because a negative times a negative is positive. So $(-3)^2 = (-3) \times (-3) = 9$.

But watch the brackets. $-3^2$ without brackets means "the negative of $3^2$", which is $-9$. The brackets change the whole answer.

So when an expression has $n^2$ and $n = -3$, substitute as $(-3)^2 = 9$.

Brackets in the expression itself also come first under BIDMAS. For $4(x + 2)$ with $x = -5$: work the bracket first, $4 \times (-5 + 2) = 4 \times (-3) = -12$. Substitute, bracket, then the rest.

A formula is a rule connecting quantities, written with letters and an equals sign. Often a real-life situation is described in words and your job is to turn it into a formula.

Example: "A plumber charges a $30 call-out fee, plus $25 for every hour worked."

1. Name the quantities. Let $C$ be the total cost and $h$ be the number of hours.
2. Describe the link in words: total cost = call-out fee + (rate per hour × hours).
3. Write it in symbols: $C = 30 + 25h$.

A few tips:

• Pick letters that hint at the quantity — $C$ for cost, $t$ for time, $h$ for hours.
• A fixed amount (the $30 here) is a number on its own.
• An amount that depends on something (the $25 per hour) is a letter multiplied by a number.

Read the formula back as a sentence to check it matches the original situation.

Once you have a formula, using it is exactly the substitution skill from earlier. Put the known numbers in place of the letters and calculate the subject.

Example: A plumber uses $C = 30 + 25h$. How much does a 4-hour job cost?

1. Write the formula: $C = 30 + 25h$.
2. Substitute the known value, $h = 4$: $C = 30 + 25 \times (4)$.
3. Apply BIDMAS — multiply first: $C = 30 + 100$.
4. Finish: $C = 130$, so the job costs $130.

The same formula handles any job. An 8-hour job: $C = 30 + 25 \times 8 = 230$, so $230. That is the power of a formula — one rule, endless answers.

Two habits keep you accurate: always write the formula down first, and always include units ($, cm, kg) in your final answer.

Getting confident with substitution and formulae now helps across many topics:

• Equations are checked by substitution, and rearranging formulae uses the same inverse skills.
• Graphs are drawn by substituting $x$-values into a rule to find $y$-values.
• Geometry and measure rely on formulae for area, perimeter, volume and angles.
• Sequences use the nth-term rule, applied by substituting the position $n$.

In everyday life, formulae are everywhere — phone bills, recipes scaled up or down, currency conversion, fuel costs and sports statistics all run on the skills in this guide.

If formula work feels hard later, it is usually a sign slip on a negative or a BIDMAS step underneath. Come back to this guide whenever you need a refresher — there is no prize for rushing.