Substitution and Formulae

Substitution is one of the most useful moves in algebra. It simply means replacing a letter with a number, then working out the answer.

Think of an expression like $n + 5$ as a vending machine slot. The letter $n$ is a gap waiting for a number. If you decide $n = 3$, you drop a 3 into the gap and the expression becomes $3 + 5 = 8$.

The method is always the same:

1. Write the expression.
2. Replace each letter with its given number.
3. Work out the answer, following BIDMAS.

One quiet trap is the hidden multiplication sign. Remember $3n$ means $3 \times n$. So if $n = 4$, then $3n = 3 \times 4 = 12$ — it is not the digits stuck together to make 34. Putting the multiplication sign back in as you substitute keeps you safe.

When an expression has several operations, the order of operations (BIDMAS) decides what you do first: Brackets, Indices, Division and Multiplication, then Addition and Subtraction.

Take the expression $2a + b$ with $a = 5$ and $b = 3$. Substitute first: $2a + b = 2 \times 5 + 3.$ Now BIDMAS: multiply before you add, so $10 + 3 = 13$.

Powers and brackets need extra care:

• For $n^2$ with $n = 4$: this is $4^2 = 4 \times 4 = 16$.
• For $3(x + 2)$ with $x = 5$: work the bracket first, $3 \times (5 + 2) = 3 \times 7 = 21$.

A neat habit is to substitute the numbers in first, writing the full sum, and only then start calculating. Rushing both steps at once is where slips creep in. Slow and tidy beats fast and wrong.

A formula is a rule that connects two or more quantities, written using letters and an equals sign. It is like a recipe — a reliable instruction you can use again and again.

You already know some formulae without thinking of them that way:

• The area of a rectangle: $A = l \times w$ (length times width).
• The perimeter of a rectangle: $P = 2(l + w)$.
• The cost of fruit: $C = 3n$, where $n$ is the number of apples at $3 each.

The single letter on its own — often on the left — is the subject of the formula. In $A = l \times w$, the subject is $A$, the quantity the formula finds for you.

Formulae are everywhere in real life: working out a phone bill, the time for a journey, the temperature in different units, or how much paint covers a wall. They let one rule handle thousands of different number combinations. Learn to read and use them, and a lot of practical maths becomes quick and easy.

Often a real-life situation is described in words, and your job is to turn it into a formula. Here is a reliable method.

Example: "A taxi charges a $4 fixed fee, plus $2 for every kilometre travelled."

1. Name the quantities with sensible letters. Let $C$ be the total cost and $k$ be the number of kilometres.
2. Describe the link in words: total cost = fixed fee + (cost per km × kilometres).
3. Write it in symbols: $C = 4 + 2k$.

A few tips:

• Pick letters that hint at the quantity — $C$ for cost, $t$ for time, $k$ for kilometres.
• A fixed amount (the $4 here) is a number on its own.
• An amount that depends on something else (the $2 per km) becomes a letter multiplied by a number.

Read your formula back as a sentence to check it makes sense. $C = 4 + 2k$ reads as "cost is 4 plus 2 for every kilometre" — exactly the original situation.

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

Example: A taxi uses the formula $C = 4 + 2k$. How much does a 7 km trip cost?

1. Write the formula: $C = 4 + 2k$.
2. Substitute the known value, $k = 7$: $C = 4 + 2 \times 7$.
3. Apply BIDMAS — multiply first: $C = 4 + 14$.
4. Finish the calculation: $C = 18$.

So the trip costs $18.

The same formula handles any trip. A 12 km trip: $C = 4 + 2 \times 12 = 28$, so $28. 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 so it means something real.

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

• Equations are solved by checking — and checking is substitution.
• 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, which you apply 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 substitution slip or a BIDMAS step underneath. Come back to this guide whenever you need a refresher — there is no prize for rushing.