1. Chapter at a Glance
"I think of a number, multiply it by 3 and add 7. I get 31. What is my number?" Puzzles like this are solved by writing an equation — a statement that two expressions are equal — and then finding the unknown value that makes it true. Equations help us solve problems about ages, money, perimeters, speeds and much more. In this chapter you will learn to form simple equations from words and solve them step by step using the idea of a balance.
- What an equation is; the difference between an expression and an equation.
- The solution of an equation and how to check it.
- Solving equations by the balance method.
- Transposing terms as a shortcut.
- Forming equations from word problems and solving them.
2. Equations and Solutions
An expression such as 3x + 7 has no "=" sign; it can take many values. An equation such as 3x + 7 = 31 says two things are equal. It is true only for particular values of the letter.
- The value of the letter that makes the equation true is its solution (or root).
- For 3x + 7 = 31, try x = 8: 24 + 7 = 31 ✔. So x = 8 is the solution.
- An equation has a left-hand side (LHS) and a right-hand side (RHS).
Trial and error works for small numbers, but a systematic method is faster and always works.
3. The Balance Method
Think of an equation as a weighing balance in perfect balance. We can keep it balanced if we do the same thing to both sides:
- add the same number to both sides;
- subtract the same number from both sides;
- multiply both sides by the same number;
- divide both sides by the same non-zero number.
Solve 3x + 7 = 31.
Solution: Subtract 7 from both sides: 3x = 24. Divide both sides by 3: x = 8. Check: 3 × 8 + 7 = 31 ✔.
Solve y/4 − 5 = 2.
Solution: Add 5: y/4 = 7. Multiply by 4: y = 28. Check: 28/4 − 5 = 7 − 5 = 2 ✔.
4. Transposing: A Shortcut
Doing the same operation on both sides has the effect of moving a term to the other side and changing its operation. This is called transposing.
| On one side | Moves to the other side as | Example |
|---|---|---|
| + a | − a | x + 9 = 15 → x = 15 − 9 = 6 |
| − a | + a | x − 4 = 10 → x = 14 |
| × a | ÷ a | 5x = 35 → x = 7 |
| ÷ a | × a | x/3 = 6 → x = 18 |
Letters on both sides
Collect letter terms on one side and numbers on the other.
Solve 5m − 3 = 2m + 12.
Solution: 5m − 2m = 12 + 3 → 3m = 15 → m = 5. Check: 22 = 22 ✔.
Solve 4(x − 2) = 2(x + 5).
Solution: 4x − 8 = 2x + 10 → 2x = 18 → x = 9.
5. Forming Equations from Word Problems
- Read carefully and decide what is unknown; call it x (or any letter).
- Write the other quantities in terms of x.
- Find the two things that are equal and write the equation.
- Solve it and check the answer in the original words.
Raju's father is 4 times as old as Raju. In 5 years, the sum of their ages will be 60. Find their present ages.
Solution: Raju = x, father = 4x. In 5 years: (x + 5) + (4x + 5) = 60 → 5x + 10 = 60 → x = 10. Raju is 10, father 40. Check: 15 + 45 = 60 ✔.
The length of a rectangle is 3 m more than twice its breadth. Its perimeter is 42 m. Find its sides.
Solution: Breadth = b, length = 2b + 3. 2(2b + 3 + b) = 42 → 6b + 6 = 42 → b = 6. Length = 15 m.
6. Solved Problems (NCERT Style)
The sum of three consecutive numbers is 75. Find them.
Solution: n + (n + 1) + (n + 2) = 75 → 3n + 3 = 75 → n = 24. Numbers: 24, 25, 26.
A purse has ₹5 and ₹2 coins, 20 coins in all, worth ₹70. How many of each?
Solution: ₹5 coins = x, ₹2 coins = 20 − x. 5x + 2(20 − x) = 70 → 3x + 40 = 70 → x = 10. 10 coins of ₹5 and 10 of ₹2.
Two angles of a triangle are equal and the third is 30° more than each of them. Find the angles.
Solution: x + x + (x + 30) = 180 → 3x = 150 → x = 50. Angles: 50°, 50°, 80°.
In a test, Sonu scored 12 marks more than half of Rinku's marks. Sonu scored 47. Find Rinku's marks.
Solution: r/2 + 12 = 47 → r/2 = 35 → r = 70.
7. Master Formula Table
| Rule | Example |
|---|---|
| Same operation on both sides keeps equality | x + 4 = 9 → x = 5 |
| +a ↔ −a when transposed | x + 6 = 10 → x = 4 |
| ×a ↔ ÷a when transposed | 7x = 56 → x = 8 |
| Collect like terms first | 2x + 3x = 25 → x = 5 |
| Consecutive numbers | n, n + 1, n + 2 |
| Consecutive odd/even | n, n + 2, n + 4 |
| Always check by substitution | LHS = RHS |
8. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Equation | Two equal expressions | Expression vs equation |
| Solution | Value making LHS = RHS | Checking answers |
| Balance method | Same operation both sides | Step-by-step solving |
| Transposing | Move and change operation | Fast solving |
| Word problems | Name unknown, form equation | Ages, coins, perimeter, angles |
9. Mind Map
- LHS = RHS
- Solution checks
- Trial and error
- Balance
- Transposing
- Letters both sides
- Brackets
- Fractions
- Ages, coins
- Perimeter, angles
10. Tips and Tricks
- Remove brackets and clear fractions first (multiply both sides by the denominator).
- Keep the letter on the side where its coefficient is larger to avoid negative coefficients.
- Always check your answer in the original problem, not just in your equation.
- Changing the sign of only one term when transposing a whole bracket.
- Dividing only one side.
- Answering x when the question asks for something else (e.g. the father's age).
11. Important One-liners
- An equation has an equals sign; an expression does not.
- The solution of an equation makes LHS equal RHS.
- Doing the same operation on both sides keeps an equation balanced.
- A term changes its operation when transposed.
- Never divide both sides by zero.
- Consecutive numbers can be written n, n + 1, n + 2.
- Always verify the solution.
12. Exam-pattern Questions with Detailed Answers
Solve x − 13 = 29.
- 16
- 42
- 32
- −16
Show answer
x = 29 + 13 = 42.
Answer: (B) 42
Solve 7p = 91.
Show answer
p = 91 ÷ 7 = 13.
Answer: 13
Is x = 4 a solution of 2x + 5 = 13?
Show answer
2 × 4 + 5 = 13 ✔.
Answer: Yes
Solve 3(a + 4) = 2a + 20.
Show answer
3a + 12 = 2a + 20 → a = 8.
Answer: a = 8
Solve (2x)/3 + 4 = 10.
Show answer
(2x)/3 = 6 → 2x = 18 → x = 9.
Answer: x = 9
I think of a number, double it and subtract 9. The result is 27. What is the number?
- 9
- 18
- 27
- 36
Show answer
2n − 9 = 27 → 2n = 36 → n = 18.
Answer: (B) 18
The perimeter of an equilateral triangle is 3 cm more than the perimeter of a square of side 5 cm. Find the side of the triangle.
Show answer
Square perimeter = 20 cm. 3s = 23 → s = 23/3 = 7⅔ cm.
Answer: 7⅔ cm
The sum of two numbers is 60 and one is 4 times the other. Find them.
Show answer
x + 4x = 60 → x = 12; other = 48.
Answer: 12 and 48
A mother is 26 years older than her daughter. In 4 years she will be three times as old as her daughter. Find their present ages.
Show answer
Daughter d, mother d + 26. In 4 years: d + 30 = 3(d + 4) → d + 30 = 3d + 12 → 2d = 18 → d = 9. Mother 35.
Check: in 4 years 13 and 39; 39 = 3 × 13 ✔.
Answer: Daughter 9, mother 35
A train ticket costs ₹85 for adults and ₹45 for children. A group of 12 paid ₹780. How many adults were there?
Show answer
85a + 45(12 − a) = 780 → 40a + 540 = 780 → a = 6.
Answer: 6 adults (and 6 children)
Solve (x − 3)/5 = (x + 1)/7.
Show answer
Multiply both sides by 35: 7(x − 3) = 5(x + 1) → 7x − 21 = 5x + 5 → 2x = 26 → x = 13.
Answer: x = 13
The digits of a two-digit number add up to 9. When the digits are reversed, the new number is 27 more than the original. Find the number.
Show answer
Tens digit t, ones digit 9 − t. Number = 10t + 9 − t = 9t + 9. Reversed = 10(9 − t) + t = 90 − 9t.
90 − 9t = 9t + 9 + 27 → 54 = 18t → t = 3. Number = 36. Check: 63 − 36 = 27 ✔.
Answer: 36
13. Quick Revision Summary
- An equation states that two expressions are equal; its solution makes both sides equal.
- Balance method: do the same to both sides.
- Transposing: +↔−, ×↔÷ when a term crosses the "=" sign.
- Remove brackets, clear fractions, collect like terms, then solve.
- Word problems: name the unknown, write the equation, solve, check in words.