Learn how variables represent changing quantities, how algebraic notation works, and how to substitute values into formulas accurately.
Use the printable version for classwork, homework or revision. It includes key ideas, practice questions and success-criteria proof.
A delivery company charges an $8 booking fee plus $1.50 for every kilometre travelled. How could you write one rule that works for any number of kilometres?
Type your first rule below. You will revisit it after learning the notation.
Write your first rule in your book. You will revisit it after learning the notation.
Core Content
A variable is a placeholder for a value that can change.
If a delivery company charges $8 plus $1.50 per kilometre, the number of kilometres will not be the same for every delivery. Algebra lets us write one rule instead of a new sentence every time.
Total cost equals 8 dollars plus $1.50 for each kilometre.
$C = 8 + 1.50k$
$C$ is the cost in dollars. $k$ is the number of kilometres.
Algebra leaves out some operation signs, but the operations are still there.
| Notation | Meaning | Example if x = 4 |
|---|---|---|
| $3x$ | $3 \times x$ | $3 \times 4 = 12$ |
| $x^2$ | $x \times x$ | $4^2 = 16$ |
| $2x + 5$ | double x, then add 5 | $2 \times 4 + 5 = 13$ |
| $\frac{x}{2}$ | x divided by 2 | $4 \div 2 = 2$ |
A delivery company uses $C = 8 + 1.50k$, where $C$ is the total cost in dollars and $k$ is the distance in kilometres. Find the cost for a 12 km delivery.
Step 1: Identify the value to substitute: $k = 12$.
Step 2: Substitute into the formula:
$C = 8 + 1.50(12)$
Step 3: Calculate in the correct order:
$C = 8 + 18 = 26$
Answer: The delivery costs $26.00.
An expression can be simplified or evaluated, but it does not make a complete statement. An equation says two quantities are equal.
| Type | Example | Why? |
|---|---|---|
| Expression | $2x + 5$ | No equals sign |
| Equation | $C = 8 + 1.50k$ | Contains an equals sign |
| Expression | $lw$ | Length times width, but not set equal to anything |
| Equation | $A = lw$ | Area equals length times width |
The perimeter of a rectangle is $P = 2l + 2w$. Find the perimeter when $l = 9$ m and $w = 4$ m.
$P = 2l + 2w$
$P = 2(9) + 2(4)$
$P = 18 + 8$
$P = 26$
Answer: The perimeter is 26 m.
When substituting a negative number, use brackets so the calculation keeps the correct meaning.
If $x = -3$, then $x^2 = (-3)^2 = 9$.
Writing $-3^2 = -9$ changes the expression because the square applies only to 3.
The delivery rule from the start can be written as $C = 8 + 1.50k$. This is more useful than a sentence because it can be reused for any distance.
Assessment
Random questions from the lesson bank - feedback appears immediately.
Write full working and interpret each answer.
1. The formula $C = 12 + 2.40m$ gives the cost in dollars for a courier trip of $m$ kilometres. Find the cost for 15 km and explain what each number in the formula means. 3 MARKS
2. A rectangle has length 11.5 cm and width 8 cm. Use $A = lw$ to find its area. Include units. 2 MARKS
3. Explain why brackets are needed when substituting $x = -4$ into $x^2 + 3x$. 3 MARKS
Test your knowledge in a rapid-fire quiz battle. Defeat the boss by answering questions correctly!