8
CBSE Marks
★★★★★
Difficulty
8
Topics
Very High
Board Weight
Topics Covered
8 key topics in this chapter
Graphical Representation
Consistent & Inconsistent Systems
Algebraic Methods: Substitution
Algebraic Methods: Elimination
Cross-Multiplication Method
Reducible Equations
Word Problems: Ages, Coins, Fractions
Speed, Distance & Work Problems
Study Resources
Key Formulas
| Formula / Rule | Expression |
|---|---|
| Unique Solution | \(a₁/a₂ ≠ b₁/b₂\) |
| Infinite Solutions | \(a₁/a₂ = b₁/b₂ = c₁/c₂\) |
| No Solution | \(a₁/a₂ = b₁/b₂ ≠ c₁/c₂\) |
| Cross-Multiplication | \(x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)\) |
Important Points to Remember
Two lines a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0 are consistent and have a unique solution when a₁/a₂ ≠ b₁/b₂.
They are consistent with infinitely many solutions when a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident lines).
They are inconsistent (no solution) when a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines).
Cross-multiplication: x/[(b₁c₂−b₂c₁)] = y/[(c₁a₂−c₂a₁)] = 1/[(a₁b₂−a₂b₁)].