Substitution and Formulae

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.

• Substitution means replacing a letter with a number.
• Wrap each value — especially negatives — in brackets.
• A negative squared is positive: $(-4)^2 = 16$.
• Follow BIDMAS once the numbers are in.

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.

• Name each quantity with a sensible letter.
• Describe the link in plain words before using symbols.
• A fixed amount is a number; a varying amount uses a letter.
• Simplify the finished formula by collecting like terms.

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.

• The subject of a formula is the letter standing alone.
• A formula tells you how to find its subject directly.
• Changing the subject rewrites the formula for a different letter.
• Rearranging uses the same inverse operations as solving equations.

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: $r = \sqrt{A}$.

• Undo each operation with its inverse, both sides equally.
• Undo $+$ and $-$ first, then $\times$ and $\div$, then powers.
• Undo a power with a root: $A = r^2$ gives $r = \sqrt{A}$.
• Finish by writing the new subject on the left.

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.

• Once a letter is the subject, finding it is just substitution.
• Rearrange once, then reuse the formula many times.
• Check by substituting your answer into the original formula.
• Include units (km, $, kg) in the final answer.

• Solving equations uses the same inverse operations as rearranging.
• Straight-line graphs are often rearranged into $y = mx + c$.
• Geometry formulae are rearranged to find missing lengths.
• Science depends on rearranging formulae constantly.

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