Linear Equations in One Variable
1. What is a Linear Equation?
A linear equation in one variable is an equation of the form ax + b = 0, where x is the variable and a, b are constants (a ≠ 0).
Key features:
- The variable has exponent 1 (no x², x³, etc.)
- Only ONE variable appears
- The highest power of the variable is 1
Examples: 2x + 5 = 13, 3y — 7 = 14, 4t + 1 = 2t — 9
2. Solving Linear Equations — Basic Rules
Rule 1 (Addition): Adding the SAME number to both sides preserves equality. Rule 2 (Subtraction): Subtracting the SAME number from both sides preserves equality. Rule 3 (Multiplication): Multiplying BOTH sides by the SAME non-zero number preserves equality. Rule 4 (Division): Dividing BOTH sides by the SAME non-zero number preserves equality.
Transposition: Moving a term from one side to the other CHANGES its sign.
3. Solving Simple Equations
Worked Example: Solve 2x + 5 = 13
2x + 5 = 13 2x = 13 — 5 (transposing +5) 2x = 8 x = 8/2 = 4
Check: 2(4) + 5 = 8 + 5 = 13 ✓
Worked Example: Solve 3(x — 2) + 4 = 2x + 7
3x — 6 + 4 = 2x + 7 3x — 2 = 2x + 7 3x — 2x = 7 + 2 x = 9
Check: 3(9—2) + 4 = 3(7) + 4 = 25. RHS: 2(9) + 7 = 25 ✓
4. Equations with Fractions
Clear denominators by multiplying BOTH sides by the LCM.
Worked Example: Solve x/3 + x/4 = 7
LCM of 3 and 4 is 12. Multiply both sides by 12: 12(x/3 + x/4) = 12 × 7 4x + 3x = 84 7x = 84 x = 12
Worked Example: Solve (2x + 1)/3 = (x — 1)/2
Cross-multiply: 2(2x + 1) = 3(x — 1) 4x + 2 = 3x — 3 4x — 3x = —3 — 2 x = —5
5. Equations Reducible to Linear Form
Worked Example: Solve (x + 2)/(x — 2) = 5/3
Cross-multiply: 3(x + 2) = 5(x — 2) 3x + 6 = 5x — 10 3x — 5x = —10 — 6 —2x = —16 x = 8
6. Word Problems — Age
Worked Example: The sum of the ages of a father and his son is 60 years. The father is three times as old as his son. Find their ages.
Let son's age = x years. Father's age = 3x years. x + 3x = 60 4x = 60 x = 15
Son is 15 years old. Father is 45 years old.
Worked Example: Five years ago, a mother was seven times as old as her son. Five years hence, she will be three times as old as her son. Find their present ages.
Let son's present age = x years. Mother's present age = y years. Five years ago: y — 5 = 7(x — 5) → y = 7x — 30 Five years hence: y + 5 = 3(x + 5) → y = 3x + 10 7x — 30 = 3x + 10 4x = 40 → x = 10 y = 3(10) + 10 = 40
Son is 10 years old. Mother is 40 years old.
7. Word Problems — Numbers
Worked Example: Find a number such that one-third of it added to one-fourth of it gives 28.
Let the number = x. x/3 + x/4 = 28 7x/12 = 28 x = 28 × 12/7 = 48
Worked Example: The sum of three consecutive integers is 51. Find them.
Let integers = x, x+1, x+2. x + (x+1) + (x+2) = 51 3x + 3 = 51 3x = 48 x = 16
Numbers: 16, 17, 18.
8. Word Problems — Money and Geometry
Worked Example: A man has Rs 500 in Rs 10 and Rs 5 notes. The number of Rs 10 notes is twice the number of Rs 5 notes. Find the number of each.
Let number of Rs 5 notes = x. Number of Rs 10 notes = 2x. 5x + 10(2x) = 500 5x + 20x = 500 25x = 500 x = 20
Rs 5 notes: 20. Rs 10 notes: 40.
Worked Example: The length of a rectangle is 3 cm more than its breadth. The perimeter is 34 cm. Find the dimensions.
Let breadth = x cm. Length = (x + 3) cm. Perimeter = 2(length + breadth) = 2(x + x + 3) = 2(2x + 3) = 34 4x + 6 = 34 4x = 28 x = 7
Breadth = 7 cm, Length = 10 cm.
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'Not changing sign when transposing' | + becomes — and — becomes + when moving across = |
| 'Multiplying only ONE term by the LCM' | Multiply EVERY term on BOTH sides by the LCM |
| 'Forgetting to check the answer' | ALWAYS substitute back into the original equation |
| 'Misreading 'less than' in word problems' | 'x is 5 less than y' means x = y — 5, NOT 5 — y |
ICSE Exam Focus (6–8 marks)
- 2-mark questions: Solve simple linear equations
- 3-mark questions: Equations with fractions or cross-multiplication
- 4-mark questions: Age or number word problems
- 6-mark questions: Multi-step word problems involving money/geometry
- 8-mark questions: Word problems reducible to linear equations
Self-Test
Q1. Solve: 5x — 3 = 2x + 9 A1. 5x — 2x = 9 + 3 → 3x = 12 → x = 4.
Q2. Solve: (x — 3)/4 + (x + 1)/5 = 2 A2. LCM = 20. 5(x—3) + 4(x+1) = 40 → 5x — 15 + 4x + 4 = 40 → 9x — 11 = 40 → 9x = 51 → x = 17/3.
Q3. The sum of two numbers is 35. Twice the larger exceeds thrice the smaller by 5. Find the numbers. A3. Let smaller = x, larger = 35 — x. 2(35—x) = 3x + 5 → 70 — 2x = 3x + 5 → 65 = 5x → x = 13. Numbers: 13 and 22.
Q4. A number plus its two-thirds gives 35. Find the number. A4. x + 2x/3 = 35 → 5x/3 = 35 → x = 21.
Q5. The ages of A and B are in the ratio 7 : 5. Ten years later, their ages will be in the ratio 9 : 7. Find their present ages. A5. Let present ages be 7x and 5x. (7x + 10)/(5x + 10) = 9/7 → 49x + 70 = 45x + 90 → 4x = 20 → x = 5. Ages: 35 years and 25 years.
Q6. The length of a rectangle is 8 m more than its breadth. If the perimeter is 64 m, find the area. A6. Let breadth = x, length = x + 8. 2(x + x + 8) = 64 → 4x + 16 = 64 → x = 12. Length = 20 m, breadth = 12 m. Area = 240 m².
