Pair of Linear Equations
Substitution, elimination, and cross-multiplication for linear systems.
Chapter Overview
A pair of linear equations in two variables can be solved graphically or algebraically. Class 10 covers substitution, elimination, cross-multiplication, and interpreting consistency (unique, infinite, or no solution).
Forms and Solutions
- General form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
- Consistent & unique: intersecting lines
- Inconsistent: parallel lines (no solution)
- Dependent: coincident lines (infinitely many solutions)
Algebraic Methods
Cross-multiplication
x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)
Condition for parallel
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Condition for coincident
a₁/a₂ = b₁/b₂ = c₁/c₂
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Elimination
Solve: x + y = 5 and x − y = 1.
- 1Add equations: 2x = 6 → x = 3
- 2Substitute: 3 + y = 5 → y = 2
Answer
x = 3, y = 2
Example 2: Word problem
Sum of two numbers is 35 and difference is 13. Find numbers.
- 1Let numbers be x, y: x + y = 35, x − y = 13
- 2Add: 2x = 48 → x = 24
- 3y = 35 − 24 = 11
Answer
24 and 11
Key Points to Remember
- ✓ Graphically: solution is point of intersection
- ✓ Check consistency before solving
- ✓ Cross-multiplication works when a₁b₂ − a₂b₁ ≠ 0
Exam Tips
- • Define variables in word problems
- • Verify solution in both equations
- • State type of solution for parallel/coincident cases
Practice MCQs — Pair of Linear Equations
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
Parallel lines have: