1. Form
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. Each is a straight line; the solution is where they meet.
2. Consistency (very important for ASO)
| Condition | Lines | Solutions |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | intersect | unique — consistent |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | coincide | infinitely many — consistent, dependent |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | parallel | none — inconsistent |
(ASO 2021) αx + y = 3 and 2x − 3y = 5 have no solution → α/2 = 1/(−3) → α = −2/3.
(ASO 2012) cx − y = 2 and 6x − 2y = 4 infinitely many → c/6 = 1/2 → c = 3.
3. Methods of solving
- Elimination: make one coefficient equal and add/subtract.
- Substitution: express one variable from one equation, put into the other.
- Cross-multiplication: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁).
- Option check: in MCQs, put each option into both equations.
2x + 3y = 13, 3x − y = 3 → from the 2nd, y = 3x − 3 → 2x + 9x − 9 = 13 → x = 2, y = 3.
4. Word problems
(ASO 2015) 4 chairs + 3 tables = ₹2100; 5 chairs + 2 tables = ₹1750 → chair ₹150, table ₹500.
Sum of digits of a 2-digit number is 9; reversing adds 27 → 9(y − x) = 27 → x = 3, y = 6 → 36.
A fraction becomes 1/2 when 1 is added to both, and 1/3 when 1 is subtracted from both → x = 3, y = 7 → 3/7.
Father's age = 3 × son's; 5 years later father = 2.5 × son's + 5 → solve with two equations.
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
Basic level
Example 1. If 4x + 11 = 39, then x = ?
(a) 7 (b) 12.5 (c) 14 (d) 8
Answer: (a) 7
Solution: 4x = 39 − 11 = 28 → x = 7.
Example 2. The sum of two numbers is 156 and their difference is 68. The larger number is:
(a) 112 (b) 44 (c) 78 (d) 113
Answer: (a) 112
Solution: Larger = (156 + 68)/2 = 112.
Example 3. The sum of two numbers is 104 and their difference is 82. The larger number is:
(a) 93 (b) 52 (c) 94 (d) 11
Answer: (a) 93
Solution: Larger = (104 + 82)/2 = 93.
ASO exam level
Example 4. 3 chairs and 2 tables cost ₹1,200, while 2 chairs and 3 tables cost ₹1,300. The cost of one chair is:
(a) ₹150 (b) ₹250 (c) ₹200 (d) ₹300
Answer: (c) ₹200
Solution: Solve the two equations: chair = ₹200, table = ₹300.
Example 5. For what value of p does the pair 4x + py + 8 = 0 and 2x + 2y + 2 = 0 have a unique solution?
(a) p ≠ 4 (b) p = 2 (c) p ≠ 2 (d) p = 4
Answer: (a) p ≠ 4
Solution: Unique solution when a₁/a₂ ≠ b₁/b₂: 4/2 ≠ p/2 → p ≠ 4.
Example 6. For what value of c do the equations cx − y = 2 and 6x − 2y − 4 = 0 have infinitely many solutions?
(a) −3 (b) 2 (c) 6 (d) 3
Answer: (d) 3
Solution: Infinite solutions when a₁/a₂ = b₁/b₂ = c₁/c₂: c/6 = −1/−2 = −2/−4 → c = 3.
Example 7. Solve: 4x + 2y = -34 and 2x − 3y = -5.
(a) x = -7, y = -3 (b) x = -7, y = -2 (c) x = 7, y = -3 (d) x = -3, y = -7
Answer: (a) x = -7, y = -3
Solution: Eliminating gives x = -7, y = -3. Check: 4(-7) + 2(-3) = -34 ✓.
Example 8. Adult tickets cost ₹21 and child tickets ₹32. 113 tickets were sold for ₹3,418. How many adult tickets were sold?
(a) 56 (b) 23 (c) 95 (d) 18
Answer: (d) 18
Solution: x + y = 113, 21x + 32y = 3418 → x = 18.
Example 9. Adult tickets cost ₹50 and child tickets ₹25. 122 tickets were sold for ₹4,175. How many adult tickets were sold?
(a) 77 (b) 50 (c) 61 (d) 45
Answer: (d) 45
Solution: x + y = 122, 50x + 25y = 4175 → x = 45.
Advanced level
Example 10. A fraction becomes 17/23 when 3 is added to both numerator and denominator, and 13/19 when 1 is subtracted from both. The fraction is:
(a) 20/14 (b) 14/20 (c) 15/20 (d) 14/21
Answer: (b) 14/20
Solution: Let it be x/y: (x + 3)/(y + 3) = 17/23, (x − 1)/(y − 1) = 13/19 → x = 14, y = 20.
Example 11. A fraction becomes 13/22 when 3 is added to both numerator and denominator, and 7/16 when 3 is subtracted from both. The fraction is:
(a) 19/10 (b) 10/20 (c) 10/19 (d) 11/19
Answer: (c) 10/19
Solution: Let it be x/y: (x + 3)/(y + 3) = 13/22, (x − 3)/(y − 3) = 7/16 → x = 10, y = 19.
Topic-wise practice for Linear equations in two variables (Quant): Solving, consistency, word problems. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Solve by substitution or elimination
- Consistent, inconsistent and dependent pairs
- Condition for unique, no or infinite solutions
- Word problems: numbers, ages, fractions, money