Pair of Linear Equations in Two Variables - Graphical Method of Solution

General Form: A pair of linear equations in two variables $x$ and $y$ is represented as $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$. Visually, each equation represents a straight line on a 2D Cartesian plane.

Geometric Interpretation: The solution to a pair of linear equations is the point of intersection of the two lines. Every point $(x, y)$ on a line satisfies the equation of that line; thus, the point where two lines meet satisfies both equations simultaneously.

Intersecting Lines (Unique Solution): If the ratio of the coefficients of $x$ and $y$ are not equal, i.e., $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the lines will cross at a single point. Visually, these lines form an 'X' shape. Such a system is called 'Consistent'.

Parallel Lines (No Solution): If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the lines have the same slope but different intercepts. Visually, they appear as two parallel lines like railway tracks that never meet. This system is 'Inconsistent'.

Coincident Lines (Infinite Solutions): If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the two equations are essentially multiples of each other. Visually, one line lies directly on top of the other, appearing as a single line. This system is 'Consistent and Dependent'.

The Table of Values: To graph a linear equation, rearrange it to the form $y = mx + c$. Choose 2 or 3 values for $x$ (e.g., $x = 0, 1, 2$) and calculate the corresponding values of $y$. These coordinates $(x, y)$ represent the points plotted to draw the straight line.

Consistency vs. Inconsistency: A pair of linear equations is consistent if it has at least one solution (intersecting or coincident lines) and inconsistent if it has no solution (parallel lines).