Hard GCSE Linear equations 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.

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 $x = 3$.

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.

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.

Subtract $4x$ from both sides

Both sides have an $x$ term. Take the smaller one, $4x$, off both sides so the unknowns end up together on the left and the coefficient stays positive.

Simplify

$6x - 4x$ leaves $2x$ on the left, and the right-hand side now has no $x$ at all.

Add $8$ to both sides

To get the $x$ term on its own, undo the $-8$ 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 $2$

The $x$ is multiplied by $2$, so the inverse operation is to divide both sides by $2$.

Simplify

$14 \div 2 = 7$, so this is the value of $x$.

Find Nia’s mistake

Nia multiplied the $3x$ by $2$ but forgot to multiply the $-4$. The number outside a bracket multiplies *everything* inside it.

Reject "$x = 5$"

Substituting $x = 5$ leaves the two sides unequal, so $x = 5$ is not a solution.

Reject "two solutions"

A linear equation (one with no $x^2$ or higher power) has exactly one solution — the $x$ terms collapse to a single term.

Reject "$10x = 14$"

Collecting $6x$ and $4x$ that sit on *opposite* sides means subtracting: $6x - 4x = 2x$, not adding them to get $10x$.

Check: substitute into the left-hand side

A solution must make the original equation true, so replace every $x$ with $7$ and work the left-hand side out.

Check: substitute into the right-hand side

Do the same on the right-hand side, again using $x = 7$.

Choose the statement

Rob is right: expanding correctly gives $6x - 8 = 4x + 6$, so $2x = 14$ and $x = 7$. Nia’s expansion of the bracket was incomplete.

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 10

The denominators are 2 and 5 and 10, and their lowest common multiple is 10. Multiply *every* term on *both* sides by 10 — miss one and the equation is no longer balanced.

Cancel the denominators

Each denominator divides exactly into 10, 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 $x$ from both sides

Both sides have an $x$ term. Take the smaller one, $x$, off both sides so the unknowns end up together on the left and the coefficient stays positive.

Simplify

$3x - x$ leaves $2x$ on the left, and the right-hand side now has no $x$ at all.

Subtract $17$ from both sides

To get the $x$ term on its own, undo the $+17$ 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 $2$

The $x$ is multiplied by $2$, so the inverse operation is to divide both sides by $2$.

Simplify

$-8 \div 2 = -4$, so this is the value of $x$.

Check: substitute into the left-hand side

A solution must make the original equation true, so replace every $x$ with $-4$ and work the left-hand side out.

Check: substitute into the right-hand side

Do the same on the right-hand side, again using $x = -4$.

Confirm the two sides balance

Both sides give the same value, so the equation balances — the solution is confirmed. This check costs seconds and catches almost every slip.

State the solution

The solution of the equation is $x = -4$.

Define the unknown

Work in pounds throughout: $5$p is $\frac{5}{100} = \frac{1}{20}$ of a pound and $9$p is $\frac{9}{100}$ of a pound.

Form the equation

Plan A costs $12 + \frac{x}{20}$ pounds and Plan B costs $8 + \frac{9x}{100}$ pounds; the two are equal.

Multiply every term by 100

The lowest common multiple of 20 and 100 is 100. Multiplying every term on both sides by 100 clears the fractions and keeps the equation balanced.

Subtract $5x$ from both sides

The right-hand side has the larger $x$ coefficient, so the unknowns are gathered there and the numbers are left on the other side.

Write the equation the other way round

An equation reads the same both ways, so swapping the sides is allowed — it just puts the unknown back on the left where we expect it.

Subtract $800$ 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 $4$

$400 \div 4 = 100$, so this is the value of $x$.

Check: substitute into the left-hand side

A solution must make the original equation true, so replace every $x$ with $100$ and work the left-hand side out.

Check: substitute into the right-hand side

Do the same on the right-hand side, again using $x = 100$.

Confirm the two sides balance

Both sides give the same value, so the equation balances — the solution is confirmed. This check costs seconds and catches almost every slip.

Interpret the answer

At $100$ minutes both plans cost $£17$. Below $100$ minutes Plan B is cheaper; above $100$ minutes Plan A is cheaper.

Watch the common error

The classic slip is to multiply only the fractions by 100 and leave the other terms alone. A whole number is a fraction over $1$, so it gets multiplied by 100 too — otherwise the two sides are no longer equal.

Alternative route

Collecting the $x$ terms on the other side instead gives the same equation the other way round, and so the same solution $x = 100$. Either route is fine — collecting on the side with the larger coefficient just avoids negative numbers.

Sense-check with a nearby value

Trying $x = 101$ makes the two sides differ, so the solution really is unique — a linear equation has exactly one solution.

State the answer

The two plans cost the same for $100$ minutes of calls per month.

Define the unknown

Choose the letter for the *smaller* quantity: if Ben is $x$ then Ali is $3x$, and no fractions appear.

Form the equation

In $5$ years Ali will be $3x + 5$ and Ben will be $x + 5$. Ali will then be twice Ben’s age, so $3x + 5 = 2(x + 5)$.

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.

Subtract $2x$ from both sides

Both sides have an $x$ term. Take the smaller one, $2x$, off both sides so the unknowns end up together on the left and the coefficient stays positive.

Simplify

$3x - 2x$ leaves $x$ on the left, and the right-hand side now has no $x$ at all.

Subtract $5$ from both sides

To get the $x$ term on its own, undo the $+5$ 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.

Work out Ali’s age

Ali is three times Ben’s age, so Ali is $15$ now.

Check: substitute into the left-hand side

A solution must make the original equation true, so replace every $x$ with $5$ and work the left-hand side out.

Check: substitute into the right-hand side

Do the same on the right-hand side, again using $x = 5$.

Confirm the two sides balance

Both sides give the same value, so the equation balances — the solution is confirmed. This check costs seconds and catches almost every slip.

Interpret the answer

In $5$ years Ali will be $20$ and Ben will be $10$ — and $20$ really is twice $10$, so the answer fits both sentences of the question.

Take care with the brackets

The number outside the bracket multiplies *everything* inside it, not just the $x$ term. If the number outside is negative, the sign of both terms inside changes.

Alternative route

Collecting the $x$ terms on the other side instead gives the same equation the other way round, and so the same solution $x = 5$. Either route is fine — collecting on the side with the larger coefficient just avoids negative numbers.

State the answer

Ben is $5$ years old now (and Ali is $15$).