Substitution and Formulae

Substitution means replacing a letter with a number, then working out the answer. By Stage 9 you handle negatives, powers and brackets together, so a steady method matters more than ever — and the safest habit is to wrap each value in brackets as you put it in.

Take the formula $A = 2b^2 - 3c$ with $b = -4$ and $c = 5$. Substitute carefully: $A = 2 \times (-4)^2 - 3 \times (5) = 2 \times 16 - 15 = 32 - 15 = 17.$

The method is always the same:

• Write the expression or formula.
• Replace each letter with its value, using brackets.
• Follow BIDMAS: brackets and powers before $\times$ and $\div$, those before $+$ and $-$.
• Remember $(-4)^2 = 16$ — a negative squared is positive.

Take it line by line and substitution stays reliable, even when negatives and powers appear together.

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

Example: "A taxi charges a $4 flag fee, plus $2 for every kilometre, then takes $5 off as a loyalty discount."

1. Name the quantities. Let $C$ be the total cost and $k$ be the number of kilometres.
2. Describe the link in words: total cost = flag fee + (rate × kilometres) − discount.
3. Write it in symbols: $C = 4 + 2k - 5$, which tidies to $C = 2k - 1$.

A few tips for Stage 9:

• Pick letters that hint at the quantity — $C$ for cost, $t$ for time.
• A fixed amount is a number on its own; an amount that depends on something is a letter times a number.
• Simplify the finished formula by collecting like terms.

Read your formula back as a sentence — it should match the original situation exactly.

Every formula has a subject — the single letter that stands alone on one side of the equals sign, with everything else on the other.

In $C = 2k - 1$, the subject is $C$: it sits by itself, and the formula tells you straight away how to find $C$ once you know $k$.

But what if you know $C$ and want $k$? The formula is the wrong way round for that. You would have to rearrange it so $k$ becomes the subject.

That is the big new Stage 9 idea: changing the subject of a formula. It means rewriting the formula so a different letter stands alone. The good news — it uses the very same inverse-operation skills you already use to solve equations. A formula is just an equation waiting to be rearranged.

To change the subject, you peel away everything around the wanted letter, using inverse operations — and you do the same thing to both sides, exactly like solving an equation.

Example: Make $k$ the subject of $C = 2k - 1$.

Step by step:

1. The $-1$ is being subtracted, so undo it by adding 1 to both sides: $C + 1 = 2k$.
2. The $k$ is being multiplied by 2, so undo it by dividing by 2: $\dfrac{C + 1}{2} = k$.
3. Write it the usual way round: $k = \dfrac{C + 1}{2}$.

The order is the reverse of BIDMAS: undo $+$ and $-$ first, then $\times$ and $\div$, then powers. If a power surrounds the letter, undo it with a root — to make $r$ the subject of $A = r^2$, take the square root: $r = \sqrt{A}$.

Why bother rearranging? Because once a letter is the subject, finding its value is plain substitution — no equation-solving needed each time.

Example: A taxi uses $C = 2k - 1$. A passenger paid $15 — how far did they travel?

Rather than solving an equation, use the rearranged version $k = \dfrac{C + 1}{2}$:

So $k = \dfrac{15 + 1}{2} = \dfrac{16}{2} = 8$ — the passenger travelled 8 km.

There is a free check built in too. Put $k = 8$ back into the original formula: $C = 2 \times 8 - 1 = 15$. It matches, so the rearrangement was correct.

Two habits keep you accurate: rearrange before substituting if you will use the formula many times, and always include units in the final answer.

Getting confident with substitution and rearranging formulae now pays off across many topics:

• Equations are solved with the very same inverse-operation moves.
• Graphs are drawn by substituting $x$-values into a rule, and lines are often rearranged into $y = mx + c$ form.
• Geometry and measure rely on formulae for area, volume and the circle — frequently rearranged to find a missing length.
• Science uses rearranged formulae everywhere, from speed and density to forces.

In everyday life, rearranging is just asking a formula a different question — a recipe scaled up, a currency conversion run backwards, a fuel-cost formula solved for distance.

If formula work feels hard later, it is usually an inverse-operation slip or a sign error underneath. Come back to this guide whenever you need a refresher — there is no prize for rushing.