Algebraic expressions

Start here: letters stand for numbers we don’t know yet

You already use letters as placeholders. “Think of a number, double it, add three” — call the number $n$ , and the instruction becomes $2n + 3$ . That’s algebra.

Three tricks that make everything easier:

$3n$ means $3 \times n$ . We drop the ”×” because it looks like an $x$ .
Like terms can be added: $3x + 2x = 5x$ , because “three of something plus two of the same something equals five of that something”.
Unlike terms cannot be combined: $3x + 2y$ stays as $3x + 2y$ — three $x$ s and two $y$ s are different things.

What you will learn

use the language of algebra: term, coefficient, variable, constant, expression,
write an expression from a word description,
collect like terms,
substitute a value into an expression,
use everyday formulas (area, sporting scores, wages, density, heart rates) and substitute real values to find unknowns,
expand a single bracket: $a(b + c) = ab + ac$ .

Worked example 0 Real-world example: AFL score formula

Collingwood kick $12$ goals and $8$ behinds. What is their score?

The AFL formula is: Score $= 6g + b$ , where $g$ = goals and $b$ = behinds.

Substitute $g = 12$ , $b = 8$ : Score $= 6(12) + 8 = 72 + 8 = 80$ .
Their opponent scores $10$ goals, $14$ behinds: $6(10) + 14 = 74$ .
Collingwood wins by $80 - 74 = 6$ points.

Key idea: the formula $6g + b$ works for every game — you never rewrite the rule, just plug in new numbers. That is why algebra uses letters.

1. The language of algebra

A pronumeral (or variable) is a letter that stands for a number. In the expression $3x + 5$ :

$3x$ and $5$ are terms (parts separated by $+$ or $-$ ),
$3$ is the coefficient of $x$ (the number multiplying the pronumeral),
$x$ is the variable,
$5$ is a constant (it does not involve a variable).

Writing expressions from words

Words	Expression
$5$ more than $n$	$n + 5$
$3$ less than $n$	$n - 3$
Twice $n$	$2n$
Half of $n$	$\dfrac{n}{2}$
Product of $a$ and $b$	$ab$
Sum of $x$ and $y$ , squared	$(x + y)^2$

2. Collecting like terms

Like terms share the same variable part. For example, $3x$ and $5x$ are like terms; $3x$ and $3y$ are not. Only like terms can be added or subtracted.

Collecting like terms

3x + 5x \;=\; 8x, \qquad 4y - y \;=\; 3y, \qquad 2a + 5 + 3a - 2 \;=\; 5a + 3.

Worked example E Very easy: collect like terms

Simplify $\;2a + 3a$ .

Both are “lots of $a$ ”. Add the coefficients: $2 + 3 = 5$ .

2a + 3a = 5a.

Another: $\;7x - 3x = 4x$ . Same idea — four $x$ s left over.

Worked example 1 Simplifying by collecting like terms

Simplify $\;7p + 3q - 2p + 5q - 4$ .

Put like terms next to each other: $\;(7p - 2p) + (3q + 5q) - 4$ .
Combine: $5p + 8q - 4$ .

3. Substitution

Substitute means “replace the pronumeral with the given number, then evaluate”.

Worked example 2 Substitution

If $a = 3$ and $b = -2$ , evaluate $\;4a - 3b + 1$ .

4(3) - 3(-2) + 1 \;=\; 12 + 6 + 1 \;=\; 19.

4. Using everyday formulas

Algebra is how we write down real-world relationships compactly. Once you know a formula, you substitute values to find the unknown.

Formulas you will meet in Year 7

Rectangle

A = L \times W \qquad \text{and} \qquad P = 2L + 2W.

$A$ is area, $L$ is length, $W$ is width, $P$ is perimeter.

AFL score

\text{total} = 6g + b.

$g$ is the number of goals (worth $6$ points each), $b$ is the number of behinds ( $1$ point each).

Weekly wage with overtime

W = b + 1.5 \times r \times h.

$b$ is the base wage for the week, $r$ is the normal hourly rate, $h$ is hours of overtime (paid at time-and-a-half).

Density

D = \dfrac{M}{V}.

$D$ is density, $M$ is mass, $V$ is volume. Common units: g/cm^3 or kg/m^3.

Maximum heart rate (rough estimate)

HR_{\max} \approx 220 - a.

$a$ is age in years. A target zone for moderate exercise is about $50\%$ to $70\%$ of $HR_{\max}$ .

Speed, distance, time

\text{speed} = \dfrac{\text{distance}}{\text{time}}, \quad \text{distance} = \text{speed} \times \text{time}.

Worked example F1 AFL score

Geelong kicked $12$ goals and $9$ behinds. Using $\text{total} = 6g + b$ , find their score.

\text{total} = 6 \times 12 + 9 = 72 + 9 = 81 \text{ points}.

Worked example F2 Weekly wage with overtime

Lara has a base wage of $400 per week and a normal hourly rate of $25. Last week she did $6$ hours of overtime. Using $W = b + 1.5rh$ , find her total pay.

W = 400 + 1.5 \times 25 \times 6 = 400 + 225 = 625.

Her total pay is $625.

Worked example F3 Density

A wooden block has mass $270$ g and volume $300$ cm^3. Using $D = \dfrac{M}{V}$ , find its density.

D = \frac{270}{300} = 0.9 \text{ g/cm}^3.

For reference, water has density $1.0$ g/cm^3, so a lower number means the block will float.

Worked example F4 Target heart rate

Using $HR_{\max} \approx 220 - a$ , find the maximum heart rate for a $13$ -year-old, and the target upper bound at $70\%$ .

Max: $HR_{\max} = 220 - 13 = 207$ beats per minute.
Upper target: $70\%$ of $207 = 0.70 \times 207 \approx 145$ bpm.

5. Expanding a single bracket

Expanding (sometimes called the distributive law) means multiplying every term inside the bracket by whatever is in front.

Distributive law

a(b + c) \;=\; ab + ac, \qquad a(b - c) \;=\; ab - ac.

Worked example 3 Expanding a bracket

Expand $\;3(2x + 5)$ .

3(2x + 5) \;=\; 3 \times 2x + 3 \times 5 \;=\; 6x + 15.

Worked example 4 Expanding and collecting

Simplify $\;5(x + 2) - 3(x - 4)$ .

Expand each bracket: $5x + 10 - 3x + 12$ . Take care with the $-3(x - 4) = -3x + 12$ .
Collect like terms: $(5x - 3x) + (10 + 12) = 2x + 22$ .

6. Simple algebraic fractions

\frac{6x + 12}{3} \;=\; \frac{6x}{3} + \frac{12}{3} \;=\; 2x + 4.

Every term in the numerator is divided by the denominator.

Practice

Fluency

Tier 1: basic skills

Write an expression for “seven more than $n$ ”.
Write an expression for “five less than $2m$ ”.
Write an expression for “the product of $4$ and $x$ ”.
Write an expression for “half of $y$ added to $3$ ”.
In the term $-7x$ , state the coefficient.
Simplify $\;4x + 3x$ .
Simplify $\;9y - 5y$ .
Simplify $\;3a + 2b + 5a - b$ .
Simplify $\;7p - 4 - 2p + 9$ .
Simplify $\;2x + 5y + 3x - y + 4$ .
Evaluate $3a + 2$ when $a = 4$ .
Evaluate $5x - y$ when $x = 2$ and $y = -3$ .
Evaluate $x^2 + 1$ when $x = -4$ .
Expand $\;2(x + 5)$ .
Expand $\;4(y - 3)$ .
Expand $\;-3(a + 2)$ .
Expand $\;6(2m - 1)$ .
Simplify $\dfrac{8x + 12}{4}$ .
Simplify $\;3(x + 2) + 5$ .
Simplify $\;2(a + 4) - 3$ .

Reasoning

Tier 2: mixed practice

Simplify $\;4(x + 3) + 2(x - 1)$ .
Simplify $\;5(2y - 1) - 3(y + 4)$ .
Simplify $\;7m + 4 - 2(m + 3)$ .
Evaluate $\;3x^2 - 2x + 1$ when $x = 4$ .
Evaluate $\;(a + b)^2$ when $a = 3$ , $b = -1$ .
Expand $\;-2(3x - 4) - (x + 2)$ .
Simplify $\dfrac{10a - 15}{5}$ .
Write an expression for the perimeter of a rectangle with length $2x + 1$ and width $x$ . Simplify it.
Write an expression for the cost of $n$ apples at $0.60 each and $m$ bananas at $0.40 each.
Find the missing coefficient: $5x + \square x = 12x$ .
Simplify $\;3a \times 4b$ .
Simplify $\;\dfrac{12xy}{3y}$ .

Reasoning

Tier 3: explain and spot the mistake

Kira writes $3 + 2x = 5x$ . Is Kira correct? If not, explain the mistake and give the correct simplification.
Explain why $3x$ and $3x^2$ are not like terms, with a numerical example.
Leo expands $-4(x - 3)$ as $-4x - 12$ . Is this right? If not, what is the correct expansion?
Write two different expressions that both equal $12$ when $x = 3$ .
Are $2(a + 3)$ and $2a + 3$ always equal? Explain with an example.

Fluency

Using everyday formulas (substitution)

Use the formulas given in the Year 7 core section.

A rectangle has $L = 12$ cm and $W = 5$ cm. Find its area and perimeter using $A = L \times W$ and $P = 2L + 2W$ .
Hawthorn scored $14$ goals and $7$ behinds. Use $\text{total} = 6g + b$ to find their total.
Collingwood kicked $9$ goals and $13$ behinds; Melbourne kicked $11$ goals and $5$ behinds. Who won, and by how much?
Sam’s base wage is $360 per week, normal rate $22/h. He worked $4$ hours of overtime. Use $W = b + 1.5rh$ to find his total pay.
A metal bar has mass $504$ g and volume $64$ cm^3. Find its density using $D = \dfrac{M}{V}$ .
Find the maximum heart rate for a person aged $45$ using $HR_{\max} = 220 - a$ .
A car travels at $80$ km/h for $2.5$ h. How far does it travel? (Use $d = s \times t$ .)
A cyclist covers $45$ km in $3$ hours. Find the average speed.
Use the formula $C = \tfrac{5}{9}(F - 32)$ to convert $68 deg$ F to degrees Celsius.

Problem solving

Tier 4: real-world problems

Mira has $x. She spends $5 on lunch and then earns $20 helping a neighbour. Write an expression for how much she has now. If she started with $12, how much has she now?
A phone plan costs a $20 monthly fee plus $0.10 per minute. Write an expression for the cost of a month with $t$ minutes of calls. What is the cost if $t = 150$ ?
The length of a rectangle is $3$ cm more than twice its width. If the width is $w$ , write expressions for the length and perimeter. Simplify the perimeter.
A taxi charges a $4.50 flag-fall plus $2 per kilometre. Write the cost for a $k$ -kilometre trip, and find the cost of a $12$ km trip.
Five students each give $x toward a gift that costs $42. Write an expression for how much change is left after the gift is bought. Evaluate it if $x = 10$ .
A swimming pool holds $V$ litres and a hose fills it at $r$ L/min. Write an expression for the time to fill the pool. How long (in minutes) if $V = 60\,000$ and $r = 150$ ?
A mobile plan charges $25 per month plus $0.08 per text. Lucy sent $t$ texts in a month. Write an expression for her total cost, then find the cost when $t = 180$ .
A gym membership costs $59 to join plus $15 per week. Write an expression for the total cost after $w$ weeks. When does the total first exceed $200?
A delivery van’s fuel cost per trip is $C = 0.15 d$ , where $d$ is the trip distance in kilometres. Find the cost of a $240$ km trip. If the fuel cost doubled per kilometre, what would the new formula be?
Daniella’s target training heart rate zone is between $50\%$ and $70\%$ of $HR_{\max} = 220 - a$ (where $a$ is her age in years). She is $12$ . Find the two ends of her target zone.