Chapter 3 · Class X Mathematics · NCERT Exercises

Pair of Linear Equations in Two Variables — Exercises

Substitution · Elimination · Cross-Multiplication — 43 Problems Solved

📂 3 Exercises 📝 12 Questions 🎓 Moderate

▶ Start Exercises 📖 Revise Notes

Exercise Index

①

Exercise 3.1

Graphical method; consistent/inconsistent/dependent pairs

②

③

3 exercise files · 12 total questions

Chapter at a Glance

CBSE BoardsNTSEOlympiad

10 Concepts

7 Formulas

Moderate Difficulty

7–9% Weightage

Before You Begin

Prerequisites

✓Linear equations in one variable (Class VIII)
✓Coordinate geometry — plotting points
✓Ratio and proportion

Have Ready

🔧Graph paper for Ex 3.1
🔧Ruler
🔧Cross-multiplication diagram

Exercise Topic Map

Exercise 3.1 Plot lines; intersecting→unique; parallel→no solution; coincident→infinite solutions

Exercise 3.2 Isolate one variable; substitute into second equation; back-substitute

Exercise 3.3 Multiply equations to equalise a coefficient; add/subtract to eliminate

Exercise 3.4 Use x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)

Exercise 3.5 Substitute u=1/x, v=1/y or similar; solve reduced pair; back-substitute

Miscellaneous Frame equations from word problem; choose most efficient method

Key Formulae — Recall Before Solving

\(\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2} \Rightarrow \text{unique solution}\)

\(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2} \Rightarrow \text{no solution}\)

\(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} \Rightarrow \text{infinitely many}\)

\(\dfrac{x}{b_1c_2-b_2c_1} = \dfrac{y}{c_1a_2-c_2a_1} = \dfrac{1}{a_1b_2-a_2b_1}\)

NCERT Solving Method

Step 1 — Classify before solving: compute a₁/a₂, b₁/b₂, c₁/c₂ ratios. Step 2 — Substitution: pick the equation with a simpler variable; isolate; substitute. Step 3 — Elimination: choose coefficients to eliminate; multiply equations as needed; solve remaining equation. Step 4 — Word problems: define variables clearly; write two distinct equations; solve; interpret and verify in context.