Expressions, Formulae and Equations | Mathematics

Unit 2 · Expressions, Formulae and Equations

2.1 Expressions, Formulae and Equations

An expression has no equals sign and nothing to solve. The moment one appears, everything changes — now there's a value hiding, and algebra is how you find it.

🎯 By the end of this you can…

✓Solve two-step linear equations by doing the same thing to both sides
✓Expand a single bracket correctly
✓Simplify expressions by collecting like terms
✓Substitute numbers into an expression or formula
✓Turn a word problem into an expression, including "X less than Y" phrasing

⚖️ Drag the +9 across the "=" to solve $4 x + 9 = 37$

=37

Drag the +9 chip past the = sign — watch it flip sign on arrival.

💡Why does crossing flip the sign? Dragging a term across the = is really subtracting it from both sides at once — it leaves one side and arrives negated on the other, so the balance never actually changes.

💡 The idea

Solving an equation: undo each operation, in reverse order

You just dragged the +9 across the equals sign and watched it flip to −9 on the other side — that's the ENTIRE method. An equation like $4 x + 9 = 37$ was BUILT by multiplying x by 4, then adding 9. To solve it, undo those steps in the opposite order: move the +9 across first, then divide by 4.

⚖️ Drag the -7 across the "=" to solve $3 x - 7 = 44$

-7

=44

Drag the -7 chip past the = sign — watch it flip sign on arrival.

🏹 Expand $5 (p - 2)$

💡 The idea

Expanding a bracket: the outside number multiplies EVERYTHING inside

You just watched both arrows fire from the 5 — one to the $p$ , one to the $- 2$ . $a (b + c) = ab + a c$ : the coefficient outside touches every single term inside the bracket, not just the first one.

🤔 A student says 5(2x + 3) = 10x + 3. What went wrong?

🔁 pq − q — change the values, watch it update live

p =q =

= 20

💡 The idea

Substitution: replace the letter with its value, then calculate normally

Try changing p and q above and watch the answer update instantly. Substitution is never more than this: put the number where the letter was, then calculate exactly like any other order-of-operations problem.

✏️ Worked example

A formula used in science is v = u + at. Find v when u = 7, a = 5 and t = 9.

1
Substitute: $v = 7 + 5 \times 9$ .
2
Multiply before adding: $5 \times 9 = 45$ .
3
Add: $7 + 45 = 52$ .

✅ Answer: $v = 52$

💡 The idea

Only MATCHING terms can combine

$p$ and $p^{2}$ are different terms, even though they share a letter — only identical terms (same letter, same power) collect together. $4 ab - 3 ba$ collects fine because $ab = ba$ , but $6 p + 3 t - 5 p + t$ can only combine the $p$ 's with each other and the $t$ 's with each other.

✏️ Worked example

Simplify 6p + 3t − 5p + t

1
Collect the $p$ terms: $6 p - 5 p = p$ .
2
Collect the $t$ terms: $3 t + t = 4 t$ .

✅ Answer: $p + 4 t$

⚠️ The classic trap

It's tempting to write $6 p + 3 t = 9 pt$ — multiplying the letters together. That's wrong: $p$ and $t$ are different unknowns being ADDED, not multiplied, and they simply can't combine into one term at all.

💡 The idea

Formulae work the same way, just with a real-world meaning attached

A bike hire formula $H = 7 + 4 t$ means: 7 is a flat starting fee, and 4 is the cost added per hour, $t$ . Substituting finds a cost; rearranging the equation finds how many hours a given cost bought.

✏️ Worked example

A school hires a bus for $180, plus $6 entry per student (n students). The total cost is $372 — how many students attended?

1
Formula: $C = 180 + 6 n$ .
2
Substitute the total: $372 = 180 + 6 n$ .
3
Subtract 180 from both sides: $6 n = 192$ .
4
Divide both sides by 6: $n = 32$ .

✅ Answer: 32 students

💡 The idea

Turning words into an expression: name the unknown first

Before writing anything, give the unknown quantity a letter. "The total cost of 7 drinks and 8 bags of chips" has TWO unknowns hiding in it — a drink's price and a bag of chips' price — so it needs two letters.

✏️ Worked example

A shop sells drinks and bags of chips. Write an expression for the total cost of 7 drinks and 8 bags of chips.

1
Let a drink cost $d$ and a bag of chips cost $c$ .
2
7 drinks cost $7 d$ ; 8 bags of chips cost $8 c$ .
3
Add the two costs together.

✅ Answer: $7 d + 8 c$

⚠️ The classic trap

"a less than 4 times b" doesn't mean $a - 4 b$ , even though $a$ is named first in the sentence. "X less than Y" always means $Y - X$ — the amount being subtracted is named FIRST in the sentence, but written SECOND in the expression: "a less than 4 times b" is $4 b - a$ .

🤔 A rectangle's length is 3 cm more than 4 times its width, w. Which expression is correct for the length?

🎮 Expressions, formulae and equations — your turn

⭐ 0 XP🔥 streak 0

1Basic · 10XP

Simplify

p + p + p + p + p

2Basic · 10XP

Write an expression for the total cost of three rings, each costing

r

3Basic · 10XP

Write an expression for "a less than 4 times b"

4Basic · 10XP

Simplify

9 p^{2} - 5 p^{2}

5Basic · 10XP

Find the value of

5 q - 1

when

q = 10

6Branching · 20XP

Simplify

4 ab - 3 ba + 7 r s - 7 sr

7Branching · 20XP

Expand

3 (5 - t)

8Branching · 20XP

Solve

7 y - 8 = 34

9Branching · 20XP

The hire cost of a bicycle is

H = 7 + 4 t

dollars for t hours. Find the cost for 5 hours.

10Challenge · 35XP

Solve

\frac{n + 3}{4} = 5

11Challenge · 35XP

An isosceles triangle has two equal sides,

4 x - 7

and 25. Find x.

12Challenge · 35XP

A triangle has sides

(2 x + 1)

(x + 4)

and

(3 x - 2)

cm. Find the perimeter when x=4.

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🏅

You finished this lesson!

Simplifying, substituting, expanding and solving are four faces of the same algebra — and every equation you'll ever meet is just "undo the steps that built it," in reverse. The Sentry is next.

Keep reading — this lesson marks itself done automatically.