Class 10 · Chapter 3

Pair of Linear Equations in Two Variables

Graphical and algebraic methods (substitution, elimination) to solve two linear equations in two variables.

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Mathematics · Class 10 (2026-27)

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Detailed theory

Each equation

a_1x + b_1y + c_1 = 0

represents a straight line. Two lines can (i) intersect at one point — unique solution (consistent, independent), (ii) be coincident — infinitely many solutions (consistent, dependent), (iii) be parallel — no solution (inconsistent). The relation between coefficients tells you which case.

Key concepts & formulas

System is consistent if it has a solution; inconsistent otherwise.

Unique solution:

\dfrac{a_1}{a_2}\ne\dfrac{b_1}{b_2}

Infinitely many:

\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}

No solution:

\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\ne\dfrac{c_1}{c_2}

Two variables need two independent equations to fix a unique point.

Graph of each equation is a straight line.