Year 9AlgebraEquations and relationshipsYear 9 Maths

Expressions, equations, and the inequality that flips

What this lesson covers

The properties of operations (commutative, distributive, associative, inverse, and identity) and the order of operations apply to numbers and variables.
When operating on or writing equations with fractions, fractions of magnitude greater than 1 are usually written as improper fractions.

Key idea

A letter is a number you have not been told yet

Algebra looks like a new subject. It is not one.
It is arithmetic done on a number whose value is blank.
Everything you know about numbers still holds.

Key idea

The same five rules, with and without numbers

Read the middle column. You agree with all of it.
The right column is the same thing with the numbers removed.
That is the whole of what algebra asks you to accept.

Property	With numbers	With letters
Commutative	7 + 3 = 3 + 7	a + b = b + a
Associative	(2 + 3) + 4 = 2 + (3 + 4)	(a + b) + c = a + (b + c)
Distributive	3 × (10 − 4) = 30 − 12	a(b − c) = ab − ac
Identity	9 + 0 = 9	a + 0 = a, and a × 1 = a
Inverse	5 + (−5) = 0	a + (−a) = 0, and a × 1/a = 1

Worked example

Collecting: add up what is the same

Three a plus five b minus a plus two b.
Gather the a terms, gather the b terms.
Two a and seven b, and they will not combine further.

Expression	Step	Gives
3a + 5b − a + 2b	the starting expression
3a − a	the a terms	2a
5b + 2b	the b terms	7b
2a + 7b	and it stops there	a and b will not combine

Key idea

Why 2a + 7b cannot be tidied further

Two apples and seven bananas is not nine of anything.
a and b stand for different unknown numbers.
Unlike terms stay separate. That is not an unfinished answer.

Worked example

Expanding: the distributive rule, forwards

Three times a bracket means three times everything inside it.
2x times, bracket, x minus 5, gives 2x squared minus 10x.
The minus sign comes through. That is the slip to watch for.

Expression	What to do	Result
3(x + 4)	3 times each thing inside	3x + 12
2x(x − 5)	2x times each thing inside	2x² − 10x
Note	the minus comes through too	not 2x² + 10x

Worked example

Factorising: the same rule, backwards

Ask what goes into every term, and take it outside.
6x plus 9: three goes into both, giving 3, bracket, 2x plus 3.
And you can always check by expanding it again.

Expression	Common factor	Factorised
6x + 9	what goes into both? 3	3(2x + 3)
4x² − 6x	what goes into both? 2x	2x(2x − 3)
Check	expand it again	you should get back

Worked example

Substitution: filling the blank in

Once you know what the letters are, put them in.
v = u + at with u five, a two, t three, gives eleven.
And the order of operations still runs. Multiply before adding.

Formula	Values	Result
Area = lw, l = 7, w = 3	7 × 3	21
v = u + at	u = 5, a = 2, t = 3	5 + 6 = 11
Order of operations	still applies, always	multiply before adding

Key idea

Solving: ask what was done to x

In 3x + 7, x was multiplied by three, then seven was added.
To get x back, undo those. In reverse order.
Subtract the seven first, then divide by the three.

Worked example

Three x plus seven equals twenty-two

Undo the add: 3x equals 15.
Undo the times: x equals 5.
Check it: three fives plus seven is twenty-two.

Line	Reasoning
3x + 7 = 22	what was done to x?	times 3, then add 7
3x = 15	undo the add 7 first
x = 5	then undo the times 3
Check	3 × 5 + 7	22, correct

Worked example

The same when there is a fraction in it

x over four, plus two, equals five. Subtract the two.
x over four is three, so x is twelve.
Nothing about the method changed.

Line	Step	Gives
x/4 + 2 = 5	subtract 2	x/4 = 3
x = 12	multiply by 4
2x/3 = 8	multiply by 3	2x = 24
x = 12	divide by 2

Worked example

Inequalities work the same way

Two x plus one is less than nine. Subtract one.
Two x is less than eight, so x is less than four.
And notice the sign did not move. We divided by a positive.

Line	Step	Note
2x + 1 < 9	subtract 1	2x < 8
x < 4	divide by 2	the sign did not move
Why not?	we divided by a positive

Key idea

But watch what a negative does

Minus three is less than five. That is true.
Multiply both sides by minus one: three, and minus five.
Three is not less than minus five. The sign has to turn round.

Statement	True?
−3 < 5	true
× both sides by −1	3 and −5
Is 3 < −5?	no, that is false
3 > −5	so the sign had to turn round

Key idea

So there is exactly one extra rule

Multiply or divide an inequality by a negative, and the sign flips.
Minus two x is greater than six gives x is less than minus three.
Adding and subtracting never flip it. Only negative times or divide.

Worked example

Rearranging is solving with letters in the answer

P equals two l plus two w. Get w on its own.
Same moves: subtract two l, then divide by two.
w is P minus two l, all over two. And then check it.

Line	Step
P = 2l + 2w	get w on its own
P − 2l = 2w	subtract 2l from both sides
(P − 2l)/2 = w	divide both sides by 2
Check: P = 20, l = 6	(20 − 12)/2	w = 4

Summary

What to take away

Six things, and the last one is the only genuinely new rule.


A letter	is a number you have not been told yet
Every arithmetic rule	works the same on letters
Expanding and factorising	are one property, both directions
Solving	is undoing, in reverse order
Rearranging	is solving with letters in the answer
Times or divide by a negative	and the inequality sign turns round

Your turn

Pause here and try all four.
Answers are in the description.

	Question
1	Simplify 5x + 3y − 2x + y.
2	Expand 4(2x − 3).
3	Solve 5x − 4 = 21.
4	Solve −2x > 6.

Show answer

1. 3x + 4y. The x terms and the y terms do not combine with each other. 2. 8x − 12. The 4 multiplies both terms, and the minus comes through. 3. x = 5. Add 4 to both sides to get 5x = 25, then divide by 5. 4. x < −3. Dividing by −2 turns the sign round. Check with x = −4: −2 × −4 is 8, and 8 is greater than 6, so it works.

Worked example

The answers

x terms and y terms do not combine.
The minus comes through the bracket.
Dividing by −2 turns the sign round.

	Answer
1	3x + 4y
2	8x − 12
3	x = 5
4	x < −3

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