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Linear equations are where algebra starts to pay off: a word problem that looks like a puzzle turns into four lines of working. They are also where small slips pile up, such as a sign lost when a term moves or a bracket only half multiplied out. This guide shows the one idea behind every method, four worked examples and a check you should never skip.
Where is this topic in the 2026-27 NCERT books?
The older Class 8 chapter "Linear Equations in One Variable" covered equations with the variable on both sides and word problems. In the 2026-27 books, the method is taught a year earlier, in Class 7 Ganita Prakash Part II, Chapter 7, "Finding the Unknown". The Class 8 books, Ganita Prakash Part I and Part II, use equations throughout but have no chapter by this name. Linear relationships return in Class 9; see our guide to linear equations in Class 9.
What does it mean to solve an equation?
"Finding the Unknown" defines an equation as a statement that two algebraic expressions are equal, and solving it as finding the value that makes the left-hand side (LHS) equal the right-hand side (RHS). The chapter starts with a weighing scale: if you take the same weight off both pans, the scale stays balanced. An equation works the same way. You can add, subtract, multiply or divide both sides by the same number (not dividing by zero), and the two sides stay equal.
The chapter also shows trial and error. For 2n + 1 = 99, trying n = 40 gives 81 and n = 50 gives 101, and n = 49 gives exactly 99. It works, but it is slow. The balance method gets there directly: subtract 1 from both sides to get 2n = 98, then divide by 2 to get n = 49.
Four worked examples
Example 1: a fraction answer (from the chapter). Solve 5x − 4 = 7.
Add 4 to both sides: 5x = 11. Divide both sides by 5: x = 11/5. Check: 5 × 11/5 − 4 = 11 − 4 = 7. Correct.
A fraction answer is fine. Students often think they have gone wrong when x is not a whole number.
Example 2: brackets and the variable on both sides. Solve 3(x − 2) = x + 8.
Expand the bracket: 3x − 6 = x + 8. Subtract x from both sides: 2x − 6 = 8. Add 6 to both sides: 2x = 14. Divide by 2: x = 7. Check: LHS = 3 × (7 − 2) = 15. RHS = 7 + 8 = 15. Correct.
Example 3: fractions in the equation. Solve x/3 + x/4 = 14.
Multiply every term by 12, the lowest common multiple of 3 and 4: 4x + 3x = 168. So 7x = 168, and x = 24. Check: 24/3 + 24/4 = 8 + 6 = 14. Correct.
Example 4: a word problem. A mother is three times as old as her daughter. In 12 years, she will be twice as old as her daughter. How old are they now?
Let the daughter's age be d. Then the mother is 3d. In 12 years: daughter d + 12, mother 3d + 12. Equation: 3d + 12 = 2(d + 12). Expand: 3d + 12 = 2d + 24. Subtract 2d: d + 12 = 24, so d = 12. The daughter is 12 and the mother is 36. Check: in 12 years they are 24 and 48, and 48 is twice 24. Correct.
What are the most common mistakes?
"Finding the Unknown" has a section called "Mind the Mistake, Mend the Mistake", and these are the errors worth hunting for in your own work:
Half-multiplied brackets: writing 3(x − 2) as 3x − 2. Every term inside gets multiplied.
Sign lost when "moving" a term: "move" is a shortcut for adding or subtracting on both sides. If you subtract x from the right, subtract it from the left too.
Clearing fractions on one term only: when you multiply by 12, multiply every term on both sides, including the 14.
Wrong unknown in word problems: write "Let d = daughter's age now" in words before starting, so you answer the question that was asked.
No check: the most costly mistake, because it lets all the others through.
How should you practise?
Retrieval works better than rereading. In Roediger and Karpicke (2006), students who took practice recall tests remembered much more on tests two days and a week later than students who reread the same material, even though rereading made them feel more confident. For equations, that means solving from a blank page, not looking at worked solutions again.
Daily five: five equations, one of each type above plus one with decimals. Check each by substitution.
Write the word problem yourself: take an equation such as 2x + 5 = 21 and invent a story it could describe. This builds the step most students find hardest.
Error audit: for every wrong answer, label the mistake from the list above.
Mix chapters: once a week, mix equations with rational number sums, since most equation errors are really fraction or sign errors.
If you are stuck on how to turn a word problem into an equation, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer. Use it after you have tried, so you can see where your set-up went differently.
Tonight, solve 4(y + 3) = 2y + 20 and check your answer by substitution.
