Pair of Linear Equations in Two Variables

What is a Pair of Linear Equations?

A pair of linear equations in two variables is a system of two equations of the form $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, where $a_1, b_1, a_2, b_2$ are real numbers (not both zero in each equation). The solution is the pair $(x, y)$ that satisfies both equations simultaneously. Geometrically, each equation represents a straight line, and the solution is the point of intersection.

Graphical Interpretation

When we plot both lines on the coordinate plane, three cases arise:

1. Intersecting lines — exactly one solution (consistent pair).
2. Parallel lines — no solution (inconsistent pair).
3. Coincident lines — infinitely many solutions (dependent, consistent pair).

The graphical method involves plotting both lines and reading the intersection point. While conceptually important, algebraic methods are faster and more accurate for exams.

Algebraic Methods of Solving

Substitution Method:
1. Express one variable in terms of the other from one equation.
2. Substitute into the second equation to get a single-variable equation.
3. Solve and back-substitute.

Elimination Method:
1. Multiply equations to make coefficients of one variable equal.
2. Add or subtract to eliminate that variable.
3. Solve the resulting equation and back-substitute.

Cross-Multiplication Method:
For $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:
$$\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}$$