Pair of Linear Equations in Two Variables – Class 10 Mathematics notes, important questions and MCQs

Chapter overview

Two linear equations in x and y are two straight lines. Their common point (if any) is the solution, found by graph, substitution or elimination.

Types of solutions

• Intersecting lines: unique solution (consistent).
• Coincident lines: infinitely many solutions (consistent, dependent).
• Parallel lines: no solution (inconsistent).

Ratio test for a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

• a₁/a₂ ≠ b₁/b₂ → unique solution.
• a₁/a₂ = b₁/b₂ = c₁/c₂ → infinitely many solutions.
• a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution.

Algebraic methods

• Substitution: express one variable from one equation and put it in the other.
• Elimination: make the coefficients of one variable equal, then add or subtract.
• Always check the answer in both equations.

Key terms

Consistent pair

a pair with at least one solution

Inconsistent pair

a pair with no solution (parallel lines)

Elimination

removing one variable by adding or subtracting the equations

Formulas

• Unique: a₁/a₂ ≠ b₁/b₂
• Infinite: a₁/a₂ = b₁/b₂ = c₁/c₂
• None: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Important questions with answers

Quick revision

• Intersecting = one solution, coincident = infinite, parallel = none.
• Use the ratio test before solving.
• Substitution and elimination give the same answer.

Exam tips

• In word problems, first write clearly what x and y stand for.
• Ratio-test questions often ask for k so that the lines are parallel — use a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Practice MCQs

1. Parallel lines represent a pair of equations with

   1. one solution
   2. no solution
   3. two solutions
   4. infinitely many solutions
   Show answer

   Answer: (b) no solution. Parallel lines never meet.

2. Solve x + y = 10, x − y = 2

   1. (4, 6)
   2. (6, 4)
   3. (8, 2)
   4. (5, 5)
   Show answer

   Answer: (b) (6, 4). Adding: 2x = 12, x = 6, y = 4.

3. Coincident lines give

   1. no solution
   2. a unique solution
   3. infinitely many solutions
   4. exactly two solutions
   Show answer

   Answer: (c) infinitely many solutions. Every point is common.

4. For 2x + 3y = 7 and 4x + 6y = 14 the pair has

   1. infinitely many solutions
   2. no solution
   3. unique solution
   4. two solutions
   Show answer

   Answer: (a) infinitely many solutions. 2/4 = 3/6 = 7/14.

5. For which k do kx + 2y = 5 and 3x + y = 1 have a unique solution?

   1. k ≠ 3
   2. k = 6
   3. k = 3
   4. k ≠ 6
   Show answer

   Answer: (d) k ≠ 6. Need k/3 ≠ 2/1, i.e. k ≠ 6.

6. The sum of two numbers is 27 and their difference is 3. The larger number is

   1. 15
   2. 12
   3. 14
   4. 13
   Show answer

   Answer: (a) 15. x + y = 27, x − y = 3 ⇒ x = 15.

7. For what k are 3x − y + 1 = 0 and 6x − 2y + k = 0 inconsistent?

   1. k = 2
   2. k = 1
   3. k ≠ 2
   4. all k
   Show answer

   Answer: (c) k ≠ 2. 3/6 = −1/−2 = 1/2; inconsistent when 1/k ≠ 1/2, i.e. k ≠ 2.

8. 5 pencils and 7 pens cost ₹50; 7 pencils and 5 pens cost ₹46. The cost of a pen is

   1. ₹3
   2. ₹6
   3. ₹4
   4. ₹5
   Show answer

   Answer: (d) ₹5. Adding: 12(x + y) = 96, x + y = 8; subtracting: 2(y − x) = 4, y − x = 2 ⇒ y = 5.