Sheet № 14 · Foundation + Higher · AQA · Edexcel · OCR
Solving Linear Equations –
Linear equations are the foundation of algebra and appear on every GCSE Maths paper, regardless of exam board or tier. A linear equation contains an unknown — usually x — raised to the power of one, and your job is to find the value of that unknown. This skill feeds directly into harder topics such as simultaneous equations, inequalities,
§Key definitions
Question:
Solve 7x − 3 = 25.
Check:
7(4) − 3 = 28 − 3 = 25 ✓
Q1 (Foundation):
Solve 4x + 9 = 37.
Q2 (Foundation/Higher):
Solve 3(2x − 5) = 4x + 7.
Q3 (Higher):
Solve (5x − 1)/3 = (2x + 4)/2.
§Formulas to memorise
If a + b = c, then a = c − b
If ax = b, then x = b ÷ a
Look at what is happening to x. — For instance, in 3x + 5 = 20, x is multiplied by 3 and then 5 is added.
Undo operations in reverse order — (think of "peeling off layers"). Subtract 5 first: 3x = 15. Then divide by 3: x = 5.
Check — by substituting back: 3(5) + 5 = 20 ✓.
Expand the brackets — first. For 4(2x − 1) = 28, expand to get 8x − 4 = 28.
Solve as a two-step equation. — Add 4: 8x = 32. Divide by 8: x = 4.
Then solve normally. — Subtract 3: 3x = 15. Divide by 3: x = 5.
Multiply every term by the lowest common denominator (LCD) — to clear the fractions.
Worked example
Solve 7x − 3 = 25.
Working:
⚠ Common mistakes
- ✗Forgetting to apply an operation to both sides. If you subtract 5 from the left, you must subtract 5 from the right too. The equals sign means both sides are balanced.
- ✗Sign errors when expanding brackets. In −2(x − 3), the answer is −2x + 6, not −2x − 6. Remember: a negative times a negative gives a positive.
- ✗Dividing only one term by the coefficient. In 3x + 6 = 21, some students divide the 6 by 3 as well. You should subtract 6 first, then divide.
- ✗Leaving fractions unfinished. If x = 15/4, write it as 3.75 or 3¾ unless the question asks for a specific form. On the calculator paper, decimals are usually fine.
- ✗Not checking the answer. A quick substitution back into the original equation catches most errors and only takes a few seconds.
✦ Exam tips
- →Show every step of working. Even if you can solve it in your head, the method marks require written steps. On AQA and Edexcel, each operation on both sides is typically worth one mark.
- →If fractions appear, clear them immediately. Multiply through by the LCD at the start — this makes the rest of the equation far simpler.
- →Circle or underline your final answer. Examiners scan quickly; make it easy for them to find x = ... in your working.
- →Practise equations with x on both sides. These are the most common mid-difficulty questions on Foundation papers and appear early on Higher papers.