One balancing method solves every linear equation. Here it is applied to all the exam types — one-step, two-step, variables on both sides, brackets, negatives and fractions — plus word problems and a full worksheet.
An equation says two things are equal. Think of it as a balance scale: whatever you do to one side, you must do to the other. Your goal is always the same — get the variable (usually x) alone on one side. This guide works through every type of linear equation an exam can set, from the simplest to fractions and word problems.
1The order that always works
- If there are fractions, multiply every term by the common denominator to clear them.
- Remove brackets by expanding.
- Get all the x-terms on one side and all the plain numbers on the other.
- Combine like terms on each side.
- Divide both sides by the number in front of x.
- Check by substituting your answer back into the original equation.
2Type 1: One-step equations
- 1Subtract 7 from both sides: x = 12 − 7.Undo the +7 to isolate x.
- 2x = 5.Check: 5 + 7 = 12 ✓
- 1Divide both sides by 3: x = 21 ÷ 3.Undo the ×3.
- 2x = 7.Check: 3 × 7 = 21 ✓
3Type 2: Two-step equations
- 1Subtract 5 from both sides: 2x = 12.Deal with the +5 first (undo addition before multiplication).
- 2Divide both sides by 2: x = 6.Undo the ×2 last.
- 3Check: 2(6) + 5 = 17 ✓Substitute back.
4Type 3: Variable on both sides
- 1Subtract 2x from both sides: 3x − 3 = 9.Gather the x-terms on the side with more x's.
- 2Add 3 to both sides: 3x = 12.Move the numbers to the other side.
- 3Divide by 3: x = 4.Check: 5(4) − 3 = 17 and 2(4) + 9 = 17 ✓
5Type 4: Equations with brackets
- 1Expand the bracket: 3x − 6 = x + 8.Multiply the 3 into both terms inside.
- 2Subtract x: 2x − 6 = 8.Gather x-terms on the left.
- 3Add 6: 2x = 14, then divide by 2: x = 7.Check: 3(7 − 2) = 15 and 7 + 8 = 15 ✓
6Type 5: Negative coefficient of x
- 1Subtract 8 from both sides: −2x = 6.Move the number away from the x-term.
- 2Divide both sides by −2: x = −3.Dividing by a negative flips the sign of the answer.
- 3Check: 8 − 2(−3) = 8 + 6 = 14 ✓Two negatives make a plus.
7Type 6: Equations with fractions
Clear the fractions first by multiplying every term by the lowest common denominator (LCD). Then solve as normal.
- 1LCD of 2 and 3 is 6. Multiply every term by 6.This removes the denominators entirely.
- 26 · x⁄2 + 6 · x⁄3 = 6 · 5 → 3x + 2x = 30.6÷2 = 3 and 6÷3 = 2.
- 3Combine: 5x = 30, so x = 6.Check: 6⁄2 + 6⁄3 = 3 + 2 = 5 ✓
8Type 7: Word problems (set up the equation)
- 1Let the number be x. Translate the words: 3x + 4 = 19.'Three times a number' = 3x; 'increased by 4' = + 4; 'is' = equals.
- 2Subtract 4: 3x = 15.Two-step method from here.
- 3Divide by 3: x = 5.The number is 5. Check: 3(5) + 4 = 19 ✓
Show every line. Markers award method marks for the balancing steps, not just the final answer — and a neat column of steps makes your own slips easy to spot.
When you move a term across the equals sign, its sign changes. Moving +6 to the other side makes it −6. Picturing 'subtract 6 from both sides' keeps the sign automatic.
This same balancing idea scales up to quadratic, simultaneous and trigonometric equations in the senior grades — so mastering these seven types now pays off for years. Try the worksheet below.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Solve x + 9 = 15 (one-step)
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x = 15 − 9 = 6 - 2Solve 4x = 32 (one-step)
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x = 32 ÷ 4 = 8 - 3Solve 3x + 7 = 22 (two-step)
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3x = 15 → x = 5 - 4Solve 5x − 4 = 21 (two-step)
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5x = 25 → x = 5 - 5Solve 6x − 5 = 2x + 11 (both sides)
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4x = 16 → x = 4 - 6Solve 7x + 3 = 4x + 18 (both sides)
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3x = 15 → x = 5 - 7Solve 2(x + 4) = 18 (brackets)
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2x + 8 = 18 → 2x = 10 → x = 5 - 8Solve 5(x − 3) = 2(x + 3) (brackets)
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5x − 15 = 2x + 6 → 3x = 21 → x = 7 - 9Solve 10 − 3x = 1 (negative x)
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−3x = −9 → x = 3 - 10Solve 12 − 5x = 2x − 2 (negative + both sides)
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12 + 2 = 2x + 5x → 14 = 7x → x = 2 - 11Solve x⁄3 + x⁄6 = 3 (fractions)
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×6: 2x + x = 18 → 3x = 18 → x = 6 - 12Solve x⁄4 − 1 = 2 (fractions)
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x⁄4 = 3 → x = 12 - 13A number doubled and then increased by 5 gives 23. Find it. (word problem)
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2x + 5 = 23 → 2x = 18 → x = 9 - 14Five less than four times a number is 27. Find it. (word problem)
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4x − 5 = 27 → 4x = 32 → x = 8
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