Key Stage 3 maths · Algebra

KS3 solving linear equations and modelling with algebra

A linear equation is a puzzle in which one unknown number, usually x, appears only to the power of one. In Key Stage 3 children learn to turn a real situation into an equation, solve it by keeping both sides balanced, cope with brackets, fractions and letters on both sides, and see what the answer means on a graph.

Solving linear equations, explained

The central idea is balance. An equation says that two things are equal, so whatever is done to one side must be done to the other. To solve 3x + 4 = 19, a child takes 4 from both sides to get 3x = 15, then divides both sides by 3 to get x = 5. Teachers often show this with a set of scales or with a bar drawn for each side, because it makes clear why the steps are allowed. Phrases such as “change sides, change sign” are shortcuts for this and are best used only after children understand the balance.

At Key Stage 3 the equations get messier, and the programme of study asks for “all forms that require rearrangement”. That includes brackets, such as 3(x − 2) = 2x + 5, which becomes 3x − 6 = 2x + 5 and then x = 11. It includes fractions, such as (x + 1)/4 = 5, where both sides are multiplied by 4 to give x + 1 = 20 and so x = 19. It also includes letters on both sides, such as 7x − 3 = 2x + 12, which gives 5x = 15 and so x = 3, and equations where the unknown is subtracted, such as 10 − x = 3, which gives x = 7. In every case the aim is the same: collect the unknowns on one side, the numbers on the other, and finish with one x.

Modelling is where the algebra gets a purpose. Children translate a story into an expression or formula: a taxi that charges £3 plus £2 per mile costs 3 + 2m for a journey of m miles. Two situations can then be compared by setting their expressions equal, which gives an equation to solve. The same model can be drawn as a graph, with the number of miles along the bottom and cost up the side. The £3 is where the line starts on the vertical axis, and the £2 per mile is how steeply it climbs.

That leads to the third idea, interpreting relationships both algebraically and graphically. Solving 30 + 20h = 10 + 25h is the same as asking where two straight lines on a graph cross. The answer for h is the horizontal position of the crossing point, and the cost at that point is its height. Children are also expected to explain what an answer means, such as why a negative number of hours makes no sense in a pricing problem, even when the algebra allows it.

Words your child will hear

Equation: A statement that two expressions are equal, such as 3x + 4 = 19.
Unknown: The letter standing for the number you are trying to find.
Inverse operation: The operation that undoes another: subtraction undoes addition, division undoes multiplication.
Linear: An equation or graph where the unknown appears only to the power of one, giving a straight line.
Model: An expression, formula or graph that describes a real situation.
Solution: The value of the unknown that makes the equation true.
Substitute to check: Put the answer back into the original equation to see whether both sides match.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs
Use algebraic methods to solve linear equations in 1 variable (including all forms that require rearrangement)
Interpret mathematical relationships both algebraically and graphically

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example: which plumber is cheaper?

Plumber A charges a £30 call-out fee plus £20 per hour. Plumber B charges a £10 call-out fee plus £25 per hour. For how many hours of work do they cost the same, and what is that cost?

Write each cost as an expression, using h for the number of hours.
```
A:  30 + 20h
B:  10 + 25h
```
The costs are the same when the two expressions are equal, so write an equation.
```
30 + 20h = 10 + 25h
```
Collect the unknowns on one side. Subtract 20h from both sides, then subtract 10 from both sides.
```
30 = 10 + 5h
20 = 5h
```
Divide both sides by 5.
```
h = 4
```
Check by substituting into both expressions: A is 30 + 20 × 4 = 110 and B is 10 + 25 × 4 = 110. They agree.
On a graph of cost against hours, A’s line starts at £30 and B’s at £10 but B is steeper, so the lines cross at (4, 110). Before 4 hours B is cheaper (at 2 hours, £60 against £70) and after 4 hours A is cheaper (at 6 hours, £150 against £160).

They cost the same for 4 hours of work, and the cost is £110.

Common mistakes and how to help

Doing the operation to only one term or one side

To solve x/3 + 2 = 5, a child multiplies both sides by 3 but only multiplies the x/3, writing x + 2 = 15 and getting x = 13. Others take a number away from one side and forget the other.

What helps: Say every step as “the same thing to the whole of both sides”. Put brackets round each side before multiplying: 3 × (x/3 + 2) = 3 × 5, which gives x + 6 = 15, so x = 9. Often it is simpler to subtract 2 first: x/3 = 3, so x = 9. Check: 9 ÷ 3 + 2 = 5.

Expanding a bracket incorrectly: 2(x + 3) becomes 2x + 3

The 2 is multiplied by the x and the multiplication is forgotten for the 3. Negative signs make it worse: 3(x − 2) becomes 3x − 2 or 3x + 6.

What helps: Draw arrows from the outside number to each term inside the bracket. Check by substituting a simple number, such as x = 1, into both the bracketed and expanded versions.

Moving a term across and forgetting that it changes sign

“Change sides, change sign” gets half-remembered, so 5x = 2x + 9 becomes 7x = 9. The rule has been applied without the balance idea behind it.

What helps: Go back to doing the same operation to both sides: take 2x from both sides to get 3x = 9. Slow and clear working is better than a fast shortcut that sometimes fails.

Rejecting an answer because it is negative or a fraction

Most worked examples have whole-number positive answers, so x = −2 or x = 3.5 feels like a mistake.

What helps: Always check by substituting. If both sides match, the answer is right. Then ask whether it makes sense in the problem: −2 hours is meaningless, but a temperature of −2 degrees is perfectly fine.

How to help at home

Think of the balance. Draw a set of balance scales. On one side put 3 bags, each holding the same unknown number of sweets, plus 4 loose sweets; on the other put 19 loose sweets. Taking 4 loose sweets off both sides keeps it level, leaving 3 bags against 15, so each bag holds 5. That is 3x + 4 = 19 solved by balancing.
Compare two deals. Two gym memberships, two phone tariffs or two taxi firms are real equations. Write each cost in letters, set them equal and find when they match. Then ask which is better for you.
Always check the answer. Make it the family rule that every solution is substituted back in. It takes ten seconds, builds the habit of testing an answer and spots most sign errors.
Make up puzzles for each other. Pick a secret number, say 6, and build an equation it solves, such as 4x − 5 = 19. Swap puzzles and solve each other’s. Writing equations teaches as much as solving them.

Builds on

Year 6: Introduction to algebra

Leads on to

Key Stage 4: Factorising, indices and identities
Key Stage 4: Straight-line and quadratic graphs
Key Stage 4: Quadratics, simultaneous equations, inequalities
Key Stage 4: Special sequences and the nth term
Key Stage 4: Algebraic fractions, proof and functions
Key Stage 4: Exponential, trig and circle graphs
Key Stage 4: Quadratic formula, iteration, inequalities
Key Stage 4: Quadratic sequences and surd ratios