Learn8300Papers 1–3 · Foundation tierAlgebraLinear and simultaneous equations

Algebra · Subtopic 3 of 9

Linear and Simultaneous Equations

An equation is a balance: whatever you do to one side, do to the other. Linear equations take a few balancing steps. Simultaneous equations have two unknowns, so you eliminate one of them first, then substitute back.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Solve linear equations, including with brackets, fractions and unknowns on both sides.
Form equations from a context and solve them.
Solve simultaneous linear equations by elimination or substitution.
Solve one linear and one quadratic equation simultaneously.

Pause and think

Two coffees and a cake cost $\$9$ ; one coffee and a cake cost $\$6$ . How much is a coffee? You probably solved this without algebra. How?

Section 2 of 14

Core idea

Start herePens and rulers

Two pens and a ruler cost 3.20; one pen and two rulers cost 2.50. One equation with two unknowns has endless answers, but two equations together pin down exactly one pair.

The big idea

Linear equations: balance. Simultaneous: eliminate one variable by adding/subtracting, or substitute.

1. Linear equations

Worked example $3(x-2)=2x+5$

Solve.

Goal: $x$ .
Strategy: Expand, collect.

$3x-6=2x+5$
Expand.
$x=11$
Subtract $2x$ , add 6.

Answer

x=11

✓ Check

3(9)=27=22+5

✓.

Worked example $\frac{x+1}3-\frac{x-2}4=1$

Solve.

Goal: $x$ .
Strategy: Multiply everything by the LCM 12.

$4(x+1)-3(x-2)=12$
×12.
$4x+4-3x+6=12$
Careful: $-3\times-2=+6$ .
$x=2$
Solve.

Answer

x=2

✓ Check

1-0=1

✓.

2. Simultaneous equations

x + y = 7 and 2x − y = 2 cross at (3, 4)

The crossing point is the one pair of values that fits both equations.

Worked example $x+y=7$ , $2x-y=2$

Solve.

Goal: $x$ , $y$ .
Strategy: $y$ coefficients are opposite: add.

$3x=9\Rightarrow x=3$
Add.
$y=7-3=4$
Substitute.

Answer

x=3

y=4

✓ Check

6-4=2

✓.

Worked example $3x+2y=16$ , $2x-5y=-2$

Solve.

Goal: $x$ , $y$ .
Strategy: Scale to match $x$ , subtract.

$6x+4y=32,\quad6x-15y=-6$
×2 and ×3.
$19y=38\Rightarrow y=2$
Subtract.
$3x+4=16\Rightarrow x=4$
Substitute.

Answer

x=4

y=2

✓ Check

8-10=-2

✓.

Worked examplePens and rulers

$2p+r=3.20$ , $p+2r=2.50$ .

Goal: Prices.
Strategy: Double the second, subtract the first.

$2p+4r=5.00$
×2.
$3r=1.80\Rightarrow r=0.60$
Subtract.
$p=2.50-1.20=1.30$
Substitute.

AnswerPen 1.30, ruler 0.60.

✓ Check

2.60+0.60=3.20

✓.

Common misconceptionSubtracting negatives

(6x+4y)-(6x-15y)=4y-(-15y)=19y

, not

-11y

Summary

Clear fractions by multiplying.
Eliminate or substitute; check in both.

Simultaneous: True at the same time.

HistoryDiophantus' age. Diophantus of Alexandria (about

250

AD) was the first to use symbols for unknowns. A puzzle said to be his epitaph: his boyhood lasted

\tfrac16

of his life, his beard grew after

\tfrac1{12}

more, he married after

\tfrac17

more, his son was born

5

years later and lived half as long as his father, who died

4

years after the son. Solving the linear equation gives

84

years.

Fun factThe Chinese elimination method. The Nine Chapters on the Mathematical Art, written in China around

200

BC, solves simultaneous equations by arranging the numbers in a grid of counting rods and eliminating, nearly

2000

years before Gauss gave the method his name in Europe.

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