Substitution

Substitution means swapping a letter in an expression or formula for a known number, then working out the value. The expression itself does not change — only the letters get replaced.

If $x = 4$, then $3x + 5$ becomes $3 \times 4 + 5 = 12 + 5 = 17$. Notice that $3x$ means "3 multiplied by $x$" — the multiplication sign is usually hidden, but it is still there once you substitute.

A small habit that pays off: always use brackets when you substitute a negative number. If $y = -3$, then $5 - 2y$ becomes $5 - 2(-3) = 5 + 6 = 11$. Without the brackets it is easy to write $5 - 2 - 3 = 0$, which is wrong. Brackets keep the sign attached to the right number.

Many expressions contain powers like $x^2$ or $x^3$. After substituting, indices come before multiplication or addition, so it really matters which step you do first.

Try $x^2 + 3x$ with $x = 5$. Substitute first: $5^2 + 3 \times 5$. Then BIDMAS: index $5^2 = 25$, multiplication $3 \times 5 = 15$, addition $25 + 15 = 40$.

Two warnings worth remembering. $(-3)^2$ is $9$, but $-3^2$ is $-9$ — because without brackets the power only attaches to the $3$ and the minus sits outside. So when you substitute $x = -3$ into $x^2$, always use brackets: $(-3)^2 = 9$. And in a product like $5xy$ with $x = 2$ and $y = 4$, the missing operation is multiplication: $5 \times 2 \times 4 = 40$.

A formula is an equation that links two or more quantities. The letter on its own — usually on the left of the equals sign — is the subject of the formula, and the others are inputs you substitute in.

Take the perimeter of a rectangle, $P = 2(l + w)$. The subject is $P$. If $l = 8$ and $w = 5$, then $P = 2(8 + 5) = 2 \times 13 = 26$. Three habits keep substitution clean: list the values first, write the formula out, then substitute.

Always include units in real-life formulae. The perimeter above would be written as $P = 26$ cm if the side lengths were in centimetres. Units make the answer mean something, and they catch silly mistakes if the numbers come out far too big or small.

Rearranging means rewriting a formula so a different letter is the subject. The rule is simple: whatever you do to one side, you do to the other — and you always use the inverse operation of what is in the way.

Inverse pairs: $+$ and $-$, $\times$ and $\div$, $x^2$ and $\sqrt{\ }$. Untie the operations in the reverse order they were tied — last on, first off.

Worked example: rearrange $v = u + at$ to make $a$ the subject.

• The $+u$ is in the way first, so subtract $u$ from both sides: $v - u = at$.
• Now $t$ is multiplying $a$, so divide both sides by $t$: $\dfrac{v - u}{t} = a$.
• Flip the equation to read with the subject on the left: $a = \dfrac{v - u}{t}$.

One step per line, one operation at a time — that is the whole technique.

Once you have an answer, take ten seconds to sense-check it. Three quick questions usually catch the slips.

• Is the size reasonable? A perimeter of $2.6$ for a room is far too small; $26$ m feels right.
• Is the sign right? Negative values turn up in temperatures, debts and depths — but a length cannot be negative.
• Are the units correct and consistent? Mixing cm and m by mistake is the classic error.

A second sense-check uses the reverse operation. If you rearranged $A = \pi r^2$ to find $r = 4$, plug $r = 4$ back into the original: $A = \pi \times 16 \approx 50.3$. Does that match the area you started with? If yes, you can move on confidently.

Sense-checking is not optional polish — it is the difference between getting full marks and almost-full marks on the value-finding questions.

Fluent substitution sits behind almost every later topic:

• Solving equations is substitution in reverse — you find the value of the letter that makes both sides agree.
• Geometry and measure use formulae for perimeter, area, volume, speed and density on every question.
• Science does the same with $F = ma$, $V = IR$ and the rest — substitute and rearrange.
• Sequences and graphs describe relationships with letters, then substitute to find specific values.

If a later topic feels slow, it is often a substitution mis-step or a rearranging slip that needs a quick refresh. Come back to this guide whenever you need a steady reset — accuracy beats speed every time.