Pair of Linear Equations in Two Variables

Equations are more important to me than politics, because politics is for the present, but equations are for eternity. — Albert Einstein

3.1 Introduction

A pair of linear equations in $x$ and $y$ has the form

$a_1 x + b_1 y + c_1 = 0,\qquad a_2 x + b_2 y + c_2 = 0$

A solution $(x,y)$ satisfies both equations.

3.2 Graphical Method

Each equation is a straight line. The solution is the point of intersection (if any).

Compare ratios $\dfrac{a_1}{a_2}$, $\dfrac{b_1}{b_2}$, $\dfrac{c_1}{c_2}$ to predict the case.

3.3 Algebraic Methods

Substitution: express one variable from one equation and substitute into the other.

Elimination: make coefficients of one variable equal (or opposite), add or subtract the equations to eliminate that variable.

Chapter Summary

• Two lines → intersecting, parallel, or coincident
• Unique solution, no solution, or infinitely many
• Solve by substitution or elimination

Exercises (NCERT)

• Exercise 3.1 — graphical method
• Exercise 3.2 — substitution
• Exercise 3.3 — elimination
• Miscellaneous Exercise