Algebraic Language, Variables and Substitution
Learn how variables represent changing quantities, how algebraic notation works, and how to substitute values into formulas accurately.
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
A delivery company charges an $8 booking fee plus $1.50 for every kilometre travelled. Without using algebrahow could you write one rule that works for any number of kilometres?
Algebra lets you write one rule instead of a new sentence every time. A variable is a placeholder for a value that can change, the same formula works for any value you substitute in.
Words: Total cost equals 8 dollars plus $1.50 for each kilometre.
Symbols: $C = 8 + 1.50k$, where $C$ is the cost in dollars and $k$ is the number of kilometres.
Key facts
- Variables represent quantities that can change.
- In algebra, $3x$ means $3 \times x$.
- An equation contains an equals sign; an expression does not.
Concepts
- A formula is a mathematical model of a relationship.
- Substitution means replacing a variable with a known value.
- Units explain what a calculated number means.
Skills
- Substitute values into formulas accurately.
- Use brackets when substituting negative values.
- Interpret the answer in the original context.
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.
$C$ is the cost in dollars. $k$ is the number of kilometres. A good formula defines what each variable means, without definitions, the symbols are easy to misread.
A variable is a letter representing a quantity that can change. Formulas use variables so one rule can solve many problems, substitute specific values to get a specific answer. Identify what each variable represents before substituting.
Pause, copy the definition of a variable (a letter representing a quantity that can change), the purpose of a formula (one rule that works for any values), and the 5-step substitution method (write formula → check units → substitute → evaluate → state answer) into your book.
Did you get this? True or false: In the formula $C = 8 + 1.50k$, the $8$ is a variable because it appears in every calculation.
We just saw that variables make formulas reusable, substitute a specific value for each variable and the formula evaluates to a specific answer. That raises a question: what does the notation in a formula actually mean, is ab a multiplication, and does a² mean something different? This card answers it → four notation rules: ab = a × b; a/b = a ÷ b; a² = a × a; 3a = 3 × a, identify the hidden operation before evaluating.
Algebra leaves out some operation signs, but the operations are still there. The table below shows what each piece of notation really means.
| 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 | $2 \times 4 + 5 = 13$ |
| $\dfrac{x}{2}$ | $x$ divided by 2 | $4 \div 2 = 2$ |
Algebraic notation: ab = a × b; a/b = a ÷ b; a² = a × a; 3a = 3 × a. These compact forms save space without losing meaning. Identify the operation hidden in each notation before evaluating.
Pause, copy the four algebraic notation rules with one example each: ab = a × b, a/b = a ÷ b, a² = a × a, 3a = 3 × a, and note that these are the operations hidden inside every formula into your book.
Quick check: If $x = 5$, what is the value of $3x + 4$?
We just saw the four algebraic notation rules that compress multiplication, division, and powers into compact symbols. That raises a question: a formula has an equals sign but a term like 3x + 2 doesn't, does that difference matter in a maths question? This card answers it → an expression is a phrase with no equals sign (can be evaluated but makes no claim); an equation has an equals sign (makes a claim that both sides are equal, and can be solved for an unknown).
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 |
Brackets matter with negative values. When substituting a negative number, use brackets so the calculation keeps the correct meaning.
If $x = -3$: correct is $x^2 = (-3)^2 = 9$. Writing $-3^2 = -9$ is wrong because the square then applies only to 3.
Expression: a mathematical phrase with no equals sign, can be evaluated but makes no claim (e.g. 3x + 2). Equation: has an equals sign, makes a claim that two sides are equal, can be solved for an unknown (e.g. 3x + 2 = 11).
Pause, copy the expression vs equation distinction: expression = no equals sign, can be evaluated (e.g. 3x + 2); equation = has equals sign, can be solved (e.g. 3x + 2 = 14), and one real-world example of each into your book.
Fill the gap: If $x = -4$, then $x^2 = ($$)^2 =$ .
Worked examples · 3 in a row, reveal as you go
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.
The perimeter of a rectangle is $P = 2l + 2w$. Find the perimeter when $l = 9$ m and $w = 4$ m.
Evaluate $x^2 + 3x$ when $x = -4$.
Match each: Drag or select the correct value for each substitution when $x = 3$.
Common errors · the 3 traps that cost marks
Quick-fire practice · 4 calculations
Evaluate $d = st$ when $s = 65$ km/h and $t = 3$ h.
Evaluate $A = lw$ when $l = 14$ cm and $w = 6$ cm.
Evaluate $T = 8 + 1.50k$ when $k = 18$.
Explain what the 8 means in $T = 8 + 1.50k$.
Teach it back: In one or two sentences, explain what substitution means and why brackets matter when substituting negative numbers.
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.
Pick your answer, then rate your confidencethat tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. 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)
Q2. A rectangle has length 11.5 cm and width 8 cm. Use $A = lw$ to find its area. Include units. (2 marks)
Q3. Explain why brackets are needed when substituting $x = -4$ into $x^2 + 3x$. (3 marks)
📖 Comprehensive answers (click to reveal)
Drill 1: $d = 65 \times 3 = 195$ km · 2: $A = 14 \times 6 = 84$ cm² · 3: $T = 8 + 1.50 \times 18 = 8 + 27 = \$35.00$ · 4: The 8 is the fixed booking fee in dollars, charged regardless of distance.
Q1 (3 marks): $C = 12 + 2.40 \times 15 = 12 + 36 = \$48.00$ [1]. The 12 is the fixed booking/base fee in dollars [1]. The 2.40 is the cost per kilometre; $m$ is the distance in kilometres [1].
Q2 (2 marks): $A = 11.5 \times 8 = 92$ cm² [2]. (Award 1 mark for correct method without units.)
Q3 (3 marks): Without brackets: $x^2 = -4^2 = -(4^2) = -16$ [1]. With brackets: $x^2 = (-4)^2 = (-4) \times (-4) = 16$ [1]. Brackets ensure the square applies to the entire negative number; without them the negative sign is not squared [1].
Five timed questions on variables, notation and substitution. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering algebra and substitution questions. Pool: lesson 1.
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