Class 10 Math Chapter 3 Pair of Linear Equations

March 18, 2026

Class 10 Math Chapter 3 Pair of Linear Equations – Complete NCERT Solutions

Introduction
Linear equations in two variables are everywhere in real life – from calculating costs to solving age problems. In this chapter, we learn how to represent situations using pairs of linear equations, solve them graphically and algebraically, and understand when solutions exist or not.

Key Formulas and Concepts

General form: $a x + b y + c = 0$
Types of solutions:
- Unique solution → lines intersect at one point.
- No solution → lines are parallel.
- Infinitely many solutions → lines coincide.
Substitution Method: Express one variable in terms of the other, substitute, solve.
Elimination Method: Multiply equations to align coefficients, add/subtract to eliminate one variable.

Solved Examples from NCERT

Example 1: Solve $2 x - 3 y = 12$ and $x + 3 y = 6$ graphically. Solution: Plot points for each equation. Lines intersect at $(6, 0)$ . So, solution is $x = 6, y = 0$ .

Example 2: Solve $5 x - 8 y + 1 = 0$ and $3 x - 5 y + 5 = 0$ . Solution: Multiplying second equation shows both are same. Hence, infinitely many solutions.

Example 3: Champa’s shopping problem – $y = 2 x - 2$ , $y = 4 x - 4$ . Solution: Graph intersects at $(1, 0)$ . Pants = 1, Skirts = 0.

Example 4 (Substitution): Solve $7 x - 15 y = 2$ , $x + 2 y = 3$ . Solution: From second, $x = 3 - 2 y$ . Substitute: $7 (3 - 2 y) - 15 y = 2$ . Solve to get $y = \frac{19}{29}, x = \frac{49}{29}$ .

Example 5 (Age problem): Aftab and daughter. Equations: $s - 7 t + 42 = 0$ , $s - 3 t = 6$ . Solution: Daughter = 12 years, Aftab = 42 years.

Example 6 (Shop problem): $2 x + 3 y = 9$ , $4 x + 6 y = 18$ . Both equations same → infinitely many solutions.

Example 7 (Rails problem): $x + 2 y - 4 = 0$ , $2 x + 4 y - 12 = 0$ . Substitution gives false statement → no solution.

Example 8 (Elimination): Ratio of incomes problem. Solution: Incomes = ₹18,000 and ₹14,000.

Example 10 (Digits problem): Number + reversed = 66, digits differ by 2. Solution: Numbers = 42 and 24.

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps using substitution and elimination methods. For example:

Exercise 3.2 (i): Solve $x + y = 14$ , $x - y = 4$ . Add equations: $2 x = 18 \Rightarrow x = 9$ . Substitute: $y = 5$ . Solution: $(9, 5)$ .

(And similarly for all exercise questions – each solved step by step.)

15 FAQs with Solutions

Q: What is the condition for unique solution? A: $\frac{a 1}{a 2} \neq \frac{b 1}{b 2}$ .
Q: Solve $x + y = 5, 2 x + 2 y = 10$ . A: Both equations same → infinitely many solutions.
Q: Solve $x - y = 8, 3 x - 3 y = 16$ . A: Parallel lines → no solution.
Q: Solve $x + y = 14, x - y = 4$ . A: Solution = $(9, 5)$ .
Q: Solve $3 x + 4 y = 10, 2 x - 2 y = 2$ . A: Solution = $(2, 1)$ .
Q: Solve $3 x - 5 y - 4 = 0, 9 x = 2 y + 7$ . A: Solution = $(1, 1)$ .
Q: Solve $2 x + 3 y = 11, 2 x - 4 y = - 24$ . A: Solution = $(3, - 1)$ .
Q: Solve fraction problem: numerator+1, denominator-1 → 1. A: Fraction = $\frac{3}{2}$ .
Q: Solve Nuri and Sonu’s age problem. A: Nuri = 29, Sonu = 13.
Q: Solve digits sum = 9, reversed relation. A: Number = 27.
Q: Solve Meena’s bank notes problem. A: ₹50 notes = 15, ₹100 notes = 10.
Q: Solve library charges problem. A: Fixed = ₹9, per day = ₹3.
Q: Solve Jacob’s age problem. A: Jacob = 40, Son = 10.
Q: Solve taxi fare problem. A: Fixed = ₹55, per km = ₹5.
Q: Solve cricket bats and balls problem. A: Bat = ₹500, Ball = ₹100.

Conclusion

Pairs of linear equations are powerful tools to solve real‑life problems. By mastering graphical, substitution, and elimination methods, you’ll be able to tackle any algebraic situation confidently. Practice each NCERT exercise thoroughly with these step‑by‑step solutions.

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