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hscscience Maths Std · Y11
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Module 1 · L1 of 13 ~45 min ⚡ +95 XP available

Algebraic Language, Variables and Substitution

Learn how variables represent changing quantities, how algebraic notation works, and how to substitute values into formulas accurately.

Today's hook, A delivery company charges $\$8$ booking fee plus $\$1.50$ per kilometre. Can one formula work for every possible trip distance?
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Worksheets

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Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

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01
Recall, your gut answer first
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A delivery company charges an $\$8$ booking fee plus $\$1.50$ for every kilometre travelled. Without using algebra how could you write one rule that works for any number of kilometres?

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The formula language you need to own
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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.

Change the input, keep the same rule, get a new output.
A variable lets one formula work for any input
Variable
A letter or symbol that represents a quantity that can change. Example: $k$ = kilometres.
Expression vs Equation
$2x + 5$ is an expression (no equals sign). $C = 8 + 1.50k$ is an equation (contains an equals sign).
Formula language
A good formula defines each variable. $d = st$ means $d$ = distance, $s$ = speed, $t$ = time.
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What you'll master
Know

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.
Understand

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.
Can do

Skills

  • Substitute values into formulas accurately.
  • Use brackets when substituting negative values.
  • Interpret the answer in the original context.
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Key terms
VariableA letter or symbol that represents a quantity that can change.
ConstantA value that stays fixed in a formula or expression.
CoefficientThe number multiplying a variable, such as 3 in $3x$.
ExpressionA mathematical phrase without an equals sign, such as $2x + 5$.
EquationA mathematical statement with an equals sign, such as $C = 8 + 1.50k$.
SubstitutionReplacing a variable with a known value.
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Variables let one rule work many times

See why a letter beats a fixed number when the same rule is reused.

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Variables let a rule work more than once
core concept

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.

$$C = 8 + 1.50k$$

$C$ 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.

Key idea. 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.

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Read and write algebraic notation

Learn the compact conventions for multiplication, division and powers.

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Algebraic notation is compact
core concept

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$4 \div 2 = 2$
Common error. $3x$ does not mean $3 + x$. It means $3 \times x$.

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 $3x + 4$?

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Tell an expression from an equation

Separate something you evaluate from something you solve.

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Expressions and equations are different
core concept

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
Communication habit. When you use a formula, write a sentence at the end. For example: "The area is 60 cm²."

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.

Exam trap. If a negative value is substituted into a power, put it in brackets first.

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 =$ .

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Work through substitution, step by step

Follow three worked substitutions, then match each step to its reason.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · SUBSTITUTE INTO A COST FORMULA

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.

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Identify: $k = 12$
The question gives the distance as 12 km. This is the value to substitute for $k$.
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PROBLEM 2 · SUBSTITUTE INTO A PERIMETER FORMULA

The perimeter of a rectangle is $P = 2l + 2w$. Find the perimeter when $l = 9$ m and $w = 4$ m.

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$P = 2l + 2w$
Write down the formula first. Identify $l = 9$ and $w = 4$.
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PROBLEM 3 · SUBSTITUTING A NEGATIVE VALUE

Evaluate $x^2 + 3x$ when $x = -4$.

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$(-4)^2 + 3(-4)$
Substitute $x = -4$ using brackets in both terms. This is critical for the power term.
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Match each: match each expression to its value when $x = 3$.

  • $2x$
  • $x^2$
  • $5x - 1$
  • $\dfrac{x}{3}$
  • 1
  • 14
  • 9
  • 6
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Avoid the traps and drill it yourself

Meet the three marks-costing errors, then run four calculations unaided.

Common errors · the 3 traps that cost marks

Trap 01
Reading $3x$ as $3 + x$
$3x$ means $3 \times x$, never $3 + x$. When $x = 4$: $3x = 12$, not 7. Juxtaposition (writing two things next to each other) always means multiplication in algebra.
Trap 02
Forgetting brackets with negatives
If $x = -3$, writing $x^2 = -3^2 = -9$ is wrong. The correct answer is $x^2 = (-3)^2 = 9$. Always use brackets when substituting negative values into any power.
Trap 03
Omitting units from the answer
$C = 26$ is incomplete. The answer is $C = \$26.00$ (dollars). Always interpret the answer in its original context and include the correct unit.

Quick-fire practice · 4 calculations

1

Evaluate $d = st$ when $s = 65$ km/h and $t = 3$ h.

2

Evaluate $A = lw$ when $l = 14$ cm and $w = 6$ cm.

3

Evaluate $T = 8 + 1.50k$ when $k = 18$.

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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.

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Revisit your first rule

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.

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Show what you have learned

Multiple choice, then short answer under exam conditions.

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

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Short answer
ApplyBand 33 marks

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)

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ApplyBand 32 marks

Q2. A rectangle has length 11.5 cm and width 8 cm. Use $A = lw$ to find its area. Include units. (2 marks)

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AnalyseBand 43 marks

Q3. Explain why brackets are needed when substituting $x = -4$ into $x^2 + 3x$. (3 marks)

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📖 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].