Solving Linear Equations Without Sign Errors

📋 Spec coverage: GCSE AQA Mathematics (8300)

• AQA A17 (Foundation), Solving linear equations: solving linear equations in one unknown algebraically, including equations with the unknown on both sides.
• This focused article does not cover equations involving brackets or finding approximate solutions using graphs.
• This skill may be assessed on Paper 1, Paper 2 or Paper 3.

Use the balance method: perform the same operation on both sides and write every step on a separate line. This is the safest answer to “How can I solve linear equations without losing track of negative signs?”

Keep the equation balanced

A linear equation is an equation in which the unknown, such as $x$, has a highest power of $1$.

Instead of thinking that a term “moves across and changes sign”, choose an inverse operation. An inverse operation reverses another operation: addition reverses subtraction, while multiplication reverses division.

For example, if one side contains $-2x$, add $2x$ to both sides. Writing the operation explicitly makes sign errors less likely.

Worked example

Solve:

$$5-2x=x-7$$

Add $2x$ to both sides:

$$5=3x-7$$

Add $7$ to both sides:

$$12=3x$$

Divide both sides by $3$:

$$x=4$$

Check by substituting $x=4$ into the original equation:

$$5-2(4)=-3$$ $$4-7=-3$$

Both sides equal $-3$, so $x=4$ is correct.

Avoid the common mistake

Do not change a sign simply because a term appears to “move”. For instance, from $-3x=12$, dividing both sides by $-3$ gives:

$$x=-4$$

A useful routine is: write the operation, apply it to both sides, simplify, then check your answer. Put negative substituted values in brackets to make their signs clear.

Completeness check: This covers algebraic linear equations with the unknown on both sides. Bracket equations and graphical approximate solutions are remaining parts of AQA A17 not covered here.