Solving by Substitution

Substituting known values into a formula and solving for the unknown, and solving a pair of equations by substituting y = … into the other.

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

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Put the Numbers In, Then Solve

Substitution means swapping a letter for what it is equal to. Here a formula gives you every value except one. Put in the numbers you know, and what is left is an equation to solve.

Worked example

1
$P = 3 a + 2 b, P = 5, b = 13$ — The question: find a. You know P and b, so a is the only unknown.
2
$5 = 3 a + 2 (13)$ — Put in the numbers you know. Use a bracket for b.
3
$5 = 3 a + 26$ — Work out 2 × 13 = 26.
4
$3 a = - 21$ — Take 26 from both sides: 5 − 26 = −21.
5
$a = - 7$ — Divide both sides by 3, because the question asks for a.

Common mistake

Stopping at 3a = −21. The question asks for a, so divide by 3.

When the Answer Is a Fraction

The answer does not have to be a whole number. A negative, a fraction or a decimal is fine.

Worked example

1
$C = 4 n + 2 k, C = 3, n = 4$ — The question: find k.
2
$3 = 4 (4) + 2 k$ — Put in the numbers you know.
3
$3 = 16 + 2 k$ — Work out 4 × 4 = 16.
4
$2 k = 3 - 16$ — Take 16 from both sides. It is 3 − 16, not 16 − 3.
5
$2 k = - 13$ — 3 − 16 = −13. The answer will be negative.
6
$k = - \frac{13}{2} = - 6.5$ — Divide both sides by 2. A fraction or a decimal is fine.

Common mistake

Working out 16 − 3 = 13. Take 16 from both sides: 3 − 16 = −13.

Two Equations: Substitute One Into the Other

With two equations, if one says $y = \dots$ , put that whole expression in place of y in the other.

Worked example

1
$y = 2 x - 1, 3 x + y = 14$ — The pair. The first one tells you exactly what y equals.
2
$3 x + (2 x - 1) = 14$ — Replace y with its whole expression, in a bracket. Now only x is left.
3
$5 x - 1 = 14$ — Collect the x terms: 3x + 2x = 5x.
4
$5 x = 15$ — Add 1 to both sides.
5
$x = 3$ — Divide both sides by 5.
6
$y = 2 (3) - 1 = 5$ — Put x = 3 into y = 2x − 1 to find y.

Key idea

Check in the other equation: $3 (3) + 5 = 14$ . Both are true, so x = 3, y = 5.

When y Has a Number in Front

Exams often say “find the point where the lines meet” (the point of intersection). That point is on both lines, so solve the two equations together. Watch out when y has a number in front.

Worked example

1
$y = 3 x - 2, x + 2 y = 17$ — The question: find where these two lines meet.
2
$x + 2 (3 x - 2) = 17$ — Replace y with 3x − 2 in a bracket. The 2 must multiply all of it.
3
$x + 6 x - 4 = 17$ — Expand: 2 × 3x = 6x and 2 × (−2) = −4.
4
$7 x - 4 = 17$ — Collect the x terms: x + 6x = 7x.
5
$7 x = 21$ — Add 4 to both sides.
6
$x = 3$ — Divide both sides by 7.
7
$y = 3 (3) - 2 = 7$ — Put x = 3 into y = 3x − 2, the easier equation.
8
$3 + 2 (7) = 17$ — Check in the other equation: 3 + 14 = 17. It works.
9
$(3, 7)$ — The question asks for a point, so write the answer as coordinates (x, y).

Common mistake

Writing 2(3x) − 2. The 2 multiplies both terms in the bracket, so it is 6x − 4.

Rearrange First, Then Substitute

Neither equation starts y = …? If one has a lone x or y (no number in front), make that letter the subject first. Then substitute as before.

Worked example

1
$x + 3 y = 11, 2 x - y = 1$ — The pair. The first one has a lone x, so it is the easy one to rearrange.
2
$x = 11 - 3 y$ — Take 3y from both sides. Now it says x = …
3
$2 (11 - 3 y) - y = 1$ — Replace x in the other equation with 11 − 3y, in a bracket.
4
$22 - 6 y - y = 1$ — Expand: 2 × 11 = 22 and 2 × (−3y) = −6y.
5
$22 - 7 y = 1$ — Collect the y terms: −6y − y = −7y.
6
$- 7 y = - 21$ — Take 22 from both sides: 1 − 22 = −21.
7
$y = 3$ — Divide both sides by −7. Minus ÷ minus is plus, so y = 3.
8
$x = 11 - 3 (3) = 2$ — Put y = 3 into x = 11 − 3y.
9
$2 (2) - 3 = 1$ — Check in the other equation: 4 − 3 = 1. It works, so x = 2, y = 3.

Common mistake

Checking in the equation you rearranged. It always fits, so it proves nothing. Check in the other one.

Which Method? Read the Question

Read the question, then pick the move.

You see	You do
A formula with one letter unknown	Put the numbers in, then solve what is left
Two equations, one starts y = … or x = …	Put that expression, in a bracket, into the other
Two equations, one has a lone x or y	Make that letter the subject, then substitute
Two equations, no lone letter	Add or subtract them to remove a letter (lesson: Simultaneous Equations by Elimination)
One equation has a squared term	Substitute the line into the curve (lesson: Simultaneous Equations with a Curve)

Key idea

Always check: put your answer back into the equation you did not use.

Key idea

Show every line, especially the one with the bracket. Most of the marks are for the working, so an answer with no working scores very little.

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Real past-paper questions on solving by substitution

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