A17, A21Linear equations

Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Algebra

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Solving equations such as \(5x - 3 = 2x + 9\), including brackets and fractions, and forming your own equation from words or a diagram (angles, perimeters, ages, costs) and solving it. It is on both tiers, from 1-mark questions to multi-step problems.

Grade by grade

What you need to be able to do, from the first marks up to the top grade.

2
Solve one-step equationsUse the inverse operation, e.g. \(x + 8 = 15\) gives \(x = 7\), and \(4x = 28\) gives \(x = 7\).
3
Solve two-step equationse.g. \(3x - 4 = 11\), so \(3x = 15\) and \(x = 5\).
4
Solve equations with bracketsExpand first, e.g. \(2(x + 5) = 18\) gives \(2x + 10 = 18\), so \(x = 4\).
4
Solve with the unknown on both sidesCollect the x terms on one side, e.g. \(7x - 3 = 4x + 12\) gives \(3x = 15\), so \(x = 5\).
5
Solve equations with a fractionMultiply both sides by the denominator, e.g. \(\frac{x + 4}{3} = 5\) gives \(x = 11\).
5
Form and solve an equation from contextWrite an equation for an angle sum, perimeter, age or cost problem, solve it and answer the question.

Notes

Balancing

An equation is like a balance: whatever you do to one side, do to the other.
Undo operations with inverses: + and −, × and ÷.
Solve \(4x + 7 = 31\): subtract 7 to get \(4x = 24\), then divide by 4 to get \(x = 6\).
Answers can be negative or fractions: \(3x = 2\) gives \(x = \frac{2}{3}\). Leave it as an exact fraction unless you are told to round.

Brackets and unknowns on both sides

Expand brackets first: \(3(2x - 1) = 21\) gives \(6x - 3 = 21\), so \(6x = 24\) and \(x = 4\).
With x on both sides, subtract the smaller x term from both sides: \(8x + 1 = 3x + 26\) gives \(5x = 25\), so \(x = 5\).
Take care with negatives: \(10 - 2x = 4\) gives \(-2x = -6\), so \(x = 3\).

Fractions

Multiply both sides by the denominator: \(\frac{x - 2}{5} = 3\) gives \(x - 2 = 15\), so \(x = 17\).

Forming equations

Choose a letter for the unknown and say what it stands for.
Use the facts in the question: angles on a straight line add to 180°, angles in a triangle add to 180°, the perimeter is the total of the sides.
Solve, then answer the actual question: if it asks for the largest angle, substitute x back in.
Check that the answer makes sense (a length cannot be negative).

Cheatsheet

Do the same to both sides
Expand brackets first
Collect x terms on one side and numbers on the other
Clear fractions by multiplying by the denominator (or the LCM of the denominators)
Angles on a straight line = 180°; in a triangle = 180°; in a quadrilateral = 360°
Check: substitute your answer into the original equation

How to answer each type of question

Solve (brackets, unknown on both sides)

3 marks4

Expand any brackets.
Collect the x terms on one side and the numbers on the other.
Divide to find x.

Example. Solve \(5(x - 2) = 2x + 11\)

Show the model answer

\(5x - 10 = 2x + 11\) M1
\(3x = 21\) M1
\(x = 7\) A1

Form and solve an equation (geometry)

3 marks5

Use the angle or perimeter fact to write an equation.
Solve it.
Substitute back to answer the question asked.

Example. The angles of a triangle are \((x + 15)°\), \(2x°\) and \((3x - 45)°\). Work out the size of the largest angle.

Show the model answer

\(x + 15 + 2x + 3x - 45 = 180\) P1
\(6x - 30 = 180\), so \(x = 35\) P1
Angles are 50°, 70° and 60°, so the largest is 70° A1

Form and solve an equation (words)

4 marks5

Write each unknown quantity in terms of one letter.
Add them (or use the given fact) to form an equation.
Solve, then give the value the question asks for.

Example. Ali is x years old. Ben is 4 years older than Ali. Cara is twice as old as Ben. Their ages add up to 60. How old is Cara?

Show the model answer

Ben is \(x + 4\) and Cara is \(2(x + 4)\) P1
\(x + x + 4 + 2x + 8 = 60\) P1
\(4x = 48\), so \(x = 12\) P1
Cara is 32 A1

Shortcuts and memory tricks

Check by substituting: for \(5(x - 2) = 2x + 11\) with \(x = 7\), both sides equal 25.
Move the smaller x term so that you are left with a positive number of x.
With fractions, clear them in the first line; after that it is an ordinary equation.

Where marks are lost

Expanding brackets wrongly, e.g. \(-3(x - 2) = -3x - 6\) (it is \(-3x + 6\)).
When multiplying by a denominator, forgetting to multiply the other terms too.
Dividing the wrong way: \(4x = 20\) gives \(x = 5\), not \(x = \frac{4}{20}\).
Stopping at the value of x when the question asks for an angle, a length or an age.

Exam technique

When the question says 'You must show all your working', an answer found by trial and improvement may get no marks.
Write the equation you formed clearly (with its total, e.g. '= 180') before solving it.
Give answers that are not whole numbers as exact fractions or decimals unless you are told to round.

Sample questions

Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.

Question 1Easy2 marks

Solve \(2(x - 5) = 6\)[2]

Show the answer and mark scheme

Answer: \(x = 8\)

M1 for \(2x - 10 = 6\) or \(x - 5 = 3\)
A1 for \(x = 8\)

Worked solution: Divide both sides by \(2\): \(x - 5 = 3\), so \(x = 8\)

Question 2Medium3 marks

Solve \(22 - 2(x + 4) = x + 11\)
You must show all your working.[3]

Show the answer and mark scheme

Answer: \(x = 1\)

M1 for expanding the bracket correctly: \(22 - 2x - 8 = x + 11\)
M1 for \(3x = 3\) oe
A1 for \(x = 1\)

Worked solution: \(14 - 2x = x + 11\)
\(3x = 3\)
\(x = 1\)

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