GCSE Linear equations Practice Questions

Write down the equation

Start from the equation as given, and keep it balanced at every stage: whatever is done to one side must be done to the other.

Subtract $7$ from both sides

To get the $x$ term on its own, undo the $+7$ by doing the opposite to both sides.

Simplify

The $x$ term is now on its own on the left; the numbers have all been gathered on the right. The solution is $x = 5$.

Form the equation

Let $x$ be the number. Multiplying by $4$ gives $4x$, adding $3$ gives $4x + 3$, and that equals $23$.

Subtract $3$ from both sides

The $x$ term is now on its own on the left; the numbers have all been gathered on the right.

Divide both sides by $4$

$20 \div 4 = 5$, so this is the value of $x$. So the number I thought of was $5$ (and $4 \times 5 + 3 = 23$, as required).

Start with the first equation

An equation has solution $x = 6$ exactly when substituting $x = 6$ makes both sides equal — so test each option in turn. Alternatively, solve each one and see which gives $6$.

Add $5$ to both sides

The $x$ term is now on its own on the left; the numbers have all been gathered on the right.

Divide both sides by $2$

$12 \div 2 = 6$, so this is the value of $x$.

Test $3x + 2 = 14$ and $\frac{x}{2} + 1 = 5$

Neither balances when $x = 6$: the first solves to $x = 4$ and the second to $x = 8$.

Test $5x = 25$ and $4x - 3 = 27$

Neither balances at $x = 6$ either: they solve to $x = 5$ and $x = 7.5$.

State the answer

Only $2x - 5 = 7$ is true when $x = 6$, so that is the correct option.

Write down the equation

Start from the equation as given, and keep it balanced at every stage: whatever is done to one side must be done to the other.

Multiply every term by 6

The lowest common multiple of 2 and 3 is 6. Multiplying every term on both sides by 6 clears the fractions and keeps the equation balanced.

Expand the brackets

Multiply everything inside each bracket by the number outside it. Take care with signs: a negative outside a bracket changes the sign of both terms inside.

Collect like terms on each side

Add together the $x$ terms, and add together the numbers, on each side separately. Nothing has crossed the equals sign yet.

Subtract $9$ from both sides

The $x$ term is now on its own on the left; the numbers have all been gathered on the right.

Compare with Sam’s Line 1

Sam multiplied every term on the *left* by $6$ but forgot the $1$ on the right. Every term on *both* sides must be multiplied — that is the first error, in Line 1.

Sam’s later lines are consistent

Lines 2, 3 and 4 follow correctly *from his wrong Line 1*, so they are not the first error.

Check Sam’s answer

His $x = -8$ does not satisfy the original equation, which confirms something went wrong.

Check the correct answer

The correct solution $x = -3$ does satisfy the original equation.

Identify the line

Line 1 contains the first error: the right-hand side was not multiplied by $6$. Correcting it gives $x = -3$.

Write down the equation

Start from the equation as given, and keep it balanced at every stage: whatever is done to one side must be done to the other.

Multiply every term by 12

The denominators are 3 and 4, and their lowest common multiple is 12. Multiply *every* term on *both* sides by 12 — miss one and the equation is no longer balanced.

Cancel the denominators

Each denominator divides exactly into 12, so the fractions disappear and only whole-number coefficients are left.

Expand the brackets

Multiply everything inside each bracket by the number outside it. Take care with signs: a negative outside a bracket changes the sign of both terms inside.

Collect like terms on each side

Add together the $x$ terms, and add together the numbers, on each side separately. Nothing has crossed the equals sign yet.

Subtract $3$ from both sides

To get the $x$ term on its own, undo the $+3$ by doing the opposite to both sides.

Simplify

The $x$ term is now on its own on the left; the numbers have all been gathered on the right.

Divide both sides by $7$

The $x$ is multiplied by $7$, so the inverse operation is to divide both sides by $7$.

Simplify

$21 \div 7 = 3$, so this is the value of $x$.

Reject "multiply only the fractions"

Every term must be multiplied — including the $2$ on the right. Leaving it as $2$ unbalances the equation and gives the wrong answer.

Reject "the $2$ has no denominator"

A whole number is a fraction with denominator $1$, so it is multiplied by $12$ just like every other term.

Reject "cancel the $x$ terms"

The $x$ terms are collected, not cancelled — they are on the same side, so they add to $7x$.

Reject "add the fractions first"

Denominators are never added. Adding the fractions *is* allowed, but it needs the common denominator $12$ — which is exactly why multiplying through by $12$ is the quicker route.

Check the solution

Put $x = 3$ back into the original equation. Both sides come to $2$, so the equation balances and the solution is right.

Choose the explanation

Multiplying every term by the lowest common multiple $12$ clears both denominators in one move and gives $4x + 3(x + 1) = 24$, which solves to $x = 3$.