POP

C2.1 Introduction to Algebra

Image · 914 × 270

01 — Explanation

Why Replace Numbers With Letters?

A letter like $x$ , $y$ , $t$ or $n$ stands in for a number that is unknown, changing, or simply "any number at all". Writing $3x+5$ instead of picking one fixed number lets a single expression describe every possible case at once — it's a shortcut for "whatever number goes in this box, do this to it". A formula does the same job for real quantities — $P=2l+2w$ works for a rectangle of any size, the moment you plug in values for $l$ and $w$ . Algebra is the language that lets a single line of working replace what would otherwise be dozens of separate arithmetic calculations.

Exam Tip

If a question gives you a real-world formula you haven't memorised (e.g. from physics or finance), don't panic — you're simply told to substitute values into it, exactly as below.

02 — Explanation

What Do "Expression", "Equation" and "Formula" Actually Mean?

These three words get mixed up constantly, but the test is simple: does it have an equals sign, and does it link real quantities?

Term	Meaning	Example
Variable	A letter representing a number that can change	$x, y, n$
Term	A number, a letter, or a product of numbers and letters	$7$ , $3x$ , $-4ab$
Expression	Terms combined with $+, -, \times, \div$ — no equals sign	$3x + 5$
Equation	Two expressions joined by an equals sign, true for particular value(s)	$3x + 5 = 20$
Formula	An equation linking two or more real-world quantities	$A = \pi r^2$

An expression can be simplified or evaluated, but it can never be "solved" — there's no equals sign to solve for. Only equations and formulas have that property, since something is being set equal to something else.

03 — Explanation

How Do I Substitute a Value Into an Expression?

Replace every occurrence of the letter with the given number, then simplify using the normal order of operations (BIDMAS) — powers before multiplication, multiplication before addition or subtraction:

\text{If } y = 3x^2 - 4, \text{ find } y \text{ when } x = 5.

y = 3(5)^2 - 4 = 3(25) - 4 = 75 - 4 = 71

04 — Explanation

How Do I Substitute a Negative Number Without Losing the Sign?

Always wrap the negative value in brackets before applying any power. This is where most marks are lost in this topic.

\text{If } y = 3x^2 - 4, \text{ find } y \text{ when } x = -2.

y = 3(-2)^2 - 4 = 3(4) - 4 = 12 - 4 = 8

Exam Tip

Compare $(-2)^2 = 4$ with the unbracketed $-2^2 = -4$ . A calculator will happily give you the wrong one if you type it without brackets — always key in $(-2)$ then square it.

05 — Explanation

How Do I Substitute Into a Formula With More Than One Letter?

Work through the letters one at a time, replacing each with its given value before combining. It helps to rewrite the formula first with gaps where each letter used to be, so nothing gets missed.

The formula for the area of a trapezium is $A = \tfrac{1}{2}(a+b)h$ , where $a$ and $b$ are the parallel sides and $h$ is the height. Find $A$ when $a=5$ , $b=9$ and $h=4$ :

A = \frac{1}{2}(5+9)(4) = \frac{1}{2}(14)(4) = \frac{1}{2}(56) = 28

Exam Tip

Substitute into the brackets first and simplify what's inside them before doing anything else — trying to multiply $a$ , $b$ and $h$ together in the wrong order is a common source of errors with multi-letter formulas.

06 — Explanation

Exam-Style Question

A plumber charges a call-out fee of $\$ 45 $plus$ \ $30$ for every hour $h$ worked.

(a) Write an expression for the total charge in dollars.

(b) Find the charge for a job that takes $3.5$ hours.

Solution

(a) The fixed part is $45$ ; the variable part is $30$ for every hour, giving $30h$ . Combine them:

\text{Charge} = 45 + 30h

(b) Substitute $h=3.5$ :

\text{Charge} = 45 + 30(3.5) = 45 + 105 = \$150

Notice part (a) asks for an expression — no equals sign, no " $=\ldots$ " at the end. Part (b) then asks for a value, which is where the substitution and the equals sign belong.