Simultaneous Equations by Elimination

Solving two linear equations by adding or subtracting, multiplying first when nothing matches, and writing equations from word problems.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1 · 106 real past-paper questions below

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What Simultaneous Equations Are

Two equations, two letters, and one pair of numbers that makes both true at the same time.

Example

$x + y = 10$ and $x - y = 4$ . Many pairs add to 10, but only $x = 7, y = 3$ also has a difference of 4.

Rule

The method: add or subtract the two equations so one letter disappears. This is called elimination.

Key idea

The answer is a pair. The question is not finished until you have both x and y.

Add or Subtract to Remove a Letter

Look for a letter with the same number in front in both equations.

Worked example

1
$3 x + 2 y = 16$ — equation (1)
2
$5 x - 2 y = 8$ — equation (2): +2y and −2y have opposite signs
3
$8 x = 24$ — add the equations: +2y and −2y make 0
4
$x = 3$ — divide both sides by 8
5
$9 + 2 y = 16$ — put x = 3 into equation (1)
6
$y = 3.5$ — 2y = 7

Opposite signs

+ 2 y and - 2 y

Add the equations.

Same signs

+ 2 y and + 2 y

Subtract the equations.

Common mistake

Stopping at x = 3. Always put it back in to find y, then check both in the other equation: $15 - 7 = 8$ .

Subtracting a Negative Term

Same signs means subtract. When one of the terms is negative, watch the signs closely.

Worked example

1
$3 x + 2 y = 10$ — Equation (1).
2
$3 x - 4 y = 16$ — Equation (2). Both have $3 x$ with the same sign, so subtract.
3
$(3 x + 2 y) - (3 x - 4 y) = 10 - 16$ — Take (2) away from (1). Brackets keep every sign safe.
4
$3 x - 3 x = 0$ — The x terms cancel. That is why we subtracted.
5
$2 y - (- 4 y) = 2 y + 4 y = 6 y$ — Taking away a negative is the same as adding, so −(−4y) becomes +4y.
6
$10 - 16 = - 6$ — Now the right side: 10 − 16 = −6.
7
$6 y = - 6$ — Put the two sides together.
8
$y = - 1$ — Divide both sides by 6.
9
$3 x + 2 (- 1) = 10$ — Put y = −1 into equation (1). Keep −1 in brackets.
10
$3 x - 2 = 10$ — 2 × (−1) = −2.
11
$3 x = 12$ — Add 2 to both sides.
12
$x = 4$ — Divide both sides by 3.
13
$12 - 4 (- 1) = 12 + 4 = 16$ — Check in equation (2). It gives 16, so x = 4, y = −1 is right.

Common mistake

Writing 2y − 4y = −2y. The minus in front of the bracket reaches the −4y too: −(−4y) = +4y.

When Nothing Matches: Multiply First

This is the most common exam question. No letter matches, so multiply the equations until one does.

Worked example

1
$3 x + 4 y = 5$ — equation (1)
2
$5 x - 3 y = 18$ — equation (2): 4y and −3y don't match
3
$9 x + 12 y = 15$ — (1) × 3: every term, the right side too
4
$20 x - 12 y = 72$ — (2) × 4: now +12y and −12y match
5
$29 x = 87$ — opposite signs, so add
6
$x = 3$ — divide both sides by 29
7
$9 + 4 y = 5$ — put x = 3 into equation (1)
8
$y = - 1$ — 4y = −4

Common mistake

Multiplying only the x and y terms. The number on the right side must be multiplied too.

Line the Equations Up First

Sometimes the letters are in a different order, or on the wrong side. Tidy both equations first, so x sits above x and y above y.

Worked example

1
$2 y + x = 3$ — Equation (1). The y comes first.
2
$3 x - 14 = 4 y$ — Equation (2). The y is on the right side.
3
$x + 2 y = 3$ — Swap the order in (1). Adding works in any order, so nothing else changes.
4
$3 x - 4 y - 14 = 0$ — In (2), take 4y from both sides.
5
$3 x - 4 y = 14$ — Add 14 to both sides. Now both equations are lined up.
6
$2 x + 4 y = 6$ — (1) × 2, every term. Now +4y and −4y match.
7
$5 x = 20$ — Opposite signs, so add: 2x + 3x = 5x and 6 + 14 = 20.
8
$x = 4$ — Divide both sides by 5.
9
$4 + 2 y = 3$ — Put x = 4 into x + 2y = 3.
10
$2 y = - 1$ — Take 4 from both sides.
11
$y = - \frac{1}{2}$ — Divide both sides by 2. A fraction is a fine answer.
12
$3 (4) - 14 = - 2, 4 (- \frac{1}{2}) = - 2$ — Check in (2) as it was given: both sides are −2.

Common mistake

Adding the equations before lining them up. Then x terms meet y terms, and nothing cancels.

Key idea

Answers like $y = - \frac{1}{2}$ or $x = 0.4$ are normal. If the check works, keep them. Don't start again.

Decimals in the Equations

Some questions use decimals. The method does not change: treat them just like whole numbers.

Worked example

1
$2 x + 3 y = 2.9$ — Equation (1).
2
$4 x + y = 2.3$ — Equation (2).
3
$4 x + 6 y = 5.8$ — (1) × 2. Double every term, the 2.9 too. Now both have 4x.
4
$5 y = 3.5$ — Same signs, so subtract (2): 6y − y = 5y and 5.8 − 2.3 = 3.5.
5
$y = 0.7$ — Divide both sides by 5.
6
$4 x + 0.7 = 2.3$ — Put y = 0.7 into equation (2).
7
$4 x = 1.6$ — Take 0.7 from both sides.
8
$x = 0.4$ — Divide both sides by 4.
9
$2 (0.4) + 3 (0.7) = 0.8 + 2.1 = 2.9$ — Check in equation (1). It gives 2.9, so it is right.

Common mistake

Forgetting to double the 2.9. When you multiply an equation, the decimal on the right is multiplied too.

A Fraction in Front (Extended/Higher)

If a letter has a fraction in front, clear the fraction first. Multiply the whole equation by the bottom number.

Worked example

1
$\frac{2 x}{3} + y = 5$ — Equation (1). The x has a fraction in front.
2
$5 x - y = 29$ — Equation (2).
3
$2 x + 3 y = 15$ — (1) × 3. The 3 on the bottom cancels. The y and the 5 are multiplied by 3 too.
4
$15 x - 3 y = 87$ — (2) × 3. Now +3y and −3y match.
5
$17 x = 102$ — Opposite signs, so add.
6
$x = 6$ — Divide both sides by 17.
7
$30 - y = 29$ — Put x = 6 into equation (2): 5 × 6 = 30.
8
$y = 1$ — 30 − 29 = 1, so y = 1.
9
$\frac{2 ( 6 )}{3} + 1 = 4 + 1 = 5$ — Check in equation (1). It gives 5, so it is right.

Common mistake

Multiplying only the fraction by 3. Every term must be multiplied, or the equation is no longer true.

Word Problems: Write Two Equations

Give each unknown price a letter. Each sentence becomes one equation.

4 pens and 3 rulers cost $5.00. 2 pens and 5 rulers cost $4.60.

Worked example

1
$4 p + 3 r = 5.00$ — p is a pen and r a ruler, in dollars
2
$2 p + 5 r = 4.60$ — the second sentence
3
$4 p + 10 r = 9.20$ — second equation × 2: now both have 4p
4
$7 r = 4.20$ — same signs, so subtract
5
$r = 0.60$ — a ruler costs $0.60
6
$4 p = 3.20, p = 0.80$ — put r back in: a pen costs $0.80

Key idea

Read the last line of the question. It may ask for a total, like 3 pens and 2 rulers: $3 (0.80) + 2 (0.60) = 3.60$ .

Not every question is about prices. With coins you need two equations: one for how many coins, one for what they are worth.

A box holds 30 coins. Each is a 5-cent or a 10-cent coin. Together they are worth $2.10.

Worked example

1
$f + t = 30$ — f five-cent coins and t ten-cent coins. This is the count equation.
2
$5 f + 10 t = 210$ — The value equation, in cents so there are no decimals: f coins at 5 cents are 5f, t coins at 10 cents are 10t, and $2.10 = 210 cents.
3
$5 f + 5 t = 150$ — Count equation × 5. Now both have 5f.
4
$5 t = 60$ — Same signs, so subtract: 10t − 5t = 5t and 210 − 150 = 60.
5
$t = 12$ — Divide both sides by 5. There are 12 ten-cent coins.
6
$f + 12 = 30$ — Put t = 12 into the count equation f + t = 30.
7
$f = 18$ — Take 12 from both sides. There are 18 five-cent coins.
8
$18 \times 5 + 12 \times 10 = 90 + 120 = 210$ — Check the value: 210 cents. It works.

Which Method? Read the Question

Line the equations up and look at the numbers in front of the letters.

You see	You do
Same number, opposite signs	Add
Same number, same signs	Subtract
Nothing matches	Multiply one or both equations first
Word problem	One letter per unknown, one equation per sentence
An $x^{2}$ or $x y$	Substitute instead: see Simultaneous Equations with a Curve
Letters out of order, or a fraction in front	Tidy first: x above x, y above y, fractions cleared

Key idea

Show every line. Label the equations (1) and (2), and write what you did, like (1) × 3. Most of the marks are for this working: an answer with no working, or found by trying numbers, scores about 1 mark. Then check both answers in the equation you did not use.

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