Solving equations by inspection and by using inverse operations — one-step, two-step, brackets and equations with exponents — plus setting up equations from word problems, with worked examples and a quiz.
An equation is a mathematical sentence with an equals sign. Solving it means finding the value of the variable that makes both sides balance. Picture a see-saw: whatever you do to one side, you must do to the other, or it tips over.
1Solving by inspection
For simple equations you can often just ask yourself the question in words.
- 1Ask: what number plus 7 gives 12?Inspection means reasoning it out.
- 2x = 5.Because 5 + 7 = 12.
2Using inverse operations
For anything harder, undo what has been done to the variable using the opposite operation. The inverse of + is −, and the inverse of × is ÷.
- 16 is being subtracted, so add 6 to both sides.Addition undoes subtraction.
- 2x − 6 + 6 = 11 + 6.Whatever you do to one side you do to the other.
- 3x = 17.Check: 17 − 6 = 11 ✓
- 1x is multiplied by 4, so divide both sides by 4.Division undoes multiplication.
- 24x ÷ 4 = 20 ÷ 4.
- 3x = 5.Check: 4 × 5 = 20 ✓
3Two-step equations
When two things have been done to the variable, undo them in reverse order — deal with the + or − first, then the × or ÷.
- 1Subtract 5 from both sides: 3x = 15.Undo the +5 first.
- 2Divide both sides by 3: x = 5.Then undo the ×3.
- 3Check: 3(5) + 5 = 15 + 5 = 20 ✓Always substitute back — it catches mistakes.
- 1Expand the bracket: 2x + 6 = 16.Or divide both sides by 2 first — either works.
- 2Subtract 6: 2x = 10.
- 3Divide by 2: x = 5.Check: 2(5+3) = 2(8) = 16 ✓
4Equations with exponents
- 1Take the square root of both sides.The inverse of squaring is the square root.
- 2x = 6 (in Grade 8 we take the positive root).Because 6 × 6 = 36.
5Setting up equations from words
- 1Let the number be x.Always name the unknown first — marks are given for this.
- 2Write the equation: 4x + 7 = 31.Translate the sentence piece by piece.
- 3Solve: 4x = 24, so x = 6.Subtract 7, then divide by 4.
- 4The number is 6.Answer the question in a sentence.
Always check by substituting your answer back into the original equation. It takes ten seconds and catches almost every arithmetic slip.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Solve: x + 9 = 15
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x = 6 - 2Solve: x − 4 = 10
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x = 14 - 3Solve: 6x = 42
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x = 7 - 4Solve: x ÷ 3 = 5
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x = 15 - 5Solve: 2x + 7 = 19
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2x = 12 → x = 6 - 6Solve: 5x − 3 = 22
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5x = 25 → x = 5 - 7Solve: 3(x − 2) = 12
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3x − 6 = 12 → 3x = 18 → x = 6 - 8Solve: x2 = 49
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x = 7 - 9A number is doubled and 5 is added, giving 17. Find the number.
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2x + 5 = 17 → 2x = 12 → x = 6
Quick Quiz
4 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 8 past papers and worksheets on this topic.