Chapter 3 Pair of Linear Equations in Two Variables

Notes

1. Linear Equation in Two Variables

A linear equation in two variables x and y is of the form:

ax+by+c=0ax + by + c = 0ax+by+c=0

where a, b, and c are real numbers and a and b are not both zero.

2. Pair of Linear Equations

A pair of linear equations in two variables is:

a1x+b1y+c1=0a_1x + b_1y + c_1 = 0a1​x+b1​y+c1​=0

a2x+b2y+c2=0a_2x + b_2y + c_2 = 0a2​x+b2​y+c2​=0

Their solution represents the point of intersection of the two straight lines.

3. Graphical Method

Each equation represents a straight line.

The point where the lines intersect gives the solution.

Types of solutions:

One solution → Lines intersect

No solution → Lines are parallel

Infinitely many solutions → Lines coincide

4. Algebraic Methods

✅ Substitution Method

Find the value of one variable from one equation

Substitute it into the other equation

✅ Elimination Method

Eliminate one variable by adding or subtracting equations

✅ Cross‑Multiplication Method

For:

a1x+b1y+c1=0a_1x + b_1y + c_1 = 0a1​x+b1​y+c1​=0

a2x+b2y+c2=0a_2x + b_2y + c_2 = 0a2​x+b2​y+c2​=0

x(b1c2−b2c1)=y(a2c1−a1c2)=1(a1b2−a2b1)\frac{x}{(b_1c_2 - b_2c_1)} =

\frac{y}{(a_2c_1 - a_1c_2)} =

\frac{1}{(a_1b_2 - a_2b_1)}(b1​c2​−b2​c1​)x​=(a2​c1​−a1​c2​)y​=(a1​b2​−a2​b1​)1