Chapter 3: Pair of Linear Equations in Two Variables
Refined and Comprehensive Class 10 Mathematics Notes
Standard Form
A pair of linear equations in two variables can be written in the general form:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
Conditions for Consistency and Solutions
Ratio Comparison
Type of Lines
Algebraic Interpretation
a₁/a₂ ≠ b₁/b₂
Intersecting lines
Unique solution (Consistent)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Parallel lines
No solution (Inconsistent)
a₁/a₂ = b₁/b₂ = c₁/c₂
Coincident lines
Infinitely many solutions (Dependent / Consistent)
Algebraic Methods of Solution
Graphical method: Plotting both equations on a graph to find point(s) of intersection.
Substitution method: Expressing one variable in terms of the other from one equation and substituting it into the second.
Elimination method: Multiplying equations by constants to make coefficients of one variable equal, then adding or subtracting.
Cross-multiplication method: Using a determinant-based formula derived from general coefficients.
Points to Remember
Consistency Check: If a system has at least one solution (unique or infinitely many), it is consistent. If it has no solution, it is inconsistent.
Word Problems Tip: Always define your variables clearly (e.g., let age of A = x years, age of B = y years) before forming the pair of linear equations.
Important Practice Questions
Solve the following pair of linear equations: 2x + 3y = 13 3x - 2y = 4
Find the value of k for which the pair of equations has infinitely many solutions.
Solve a word problem involving ages, numbers, or prices using two linear equations.
Quick Revision
Unique Solution
a₁/a₂ ≠ b₁/b₂ (Lines intersect at a single point)
No Solution
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (Lines are parallel)
Infinitely Many Solutions
a₁/a₂ = b₁/b₂ = c₁/c₂ (Lines overlap entirely)
Main Methods
Substitution, Elimination, Graphical, and Cross-multiplication.