Pair of Linear Equations in Two Variables - Consistency and Inconsistency of Solutions

General Form: A pair of linear equations in two variables $x$ and $y$ is typically written as $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$. Here, $a_1, b_1, c_1, a_2, b_2, c_2$ are real numbers such that $a_1^2 + b_1^2 \neq 0$ and $a_2^2 + b_2^2 \neq 0$.

Consistent System: A system is called consistent if it has at least one solution. Visually, this is represented by lines that either cross at a single point or overlap entirely, meaning there is at least one set of coordinates $(x, y)$ that satisfies both equations.

Inconsistent System: A system is called inconsistent if it has no solution. In a coordinate plane, this is visually depicted as two parallel lines that never intersect, no matter how far they are extended, because they share no common points.

Unique Solution (Intersecting Lines): If the ratio of the coefficients of $x$ is not equal to the ratio of the coefficients of $y$ (i.e., $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$), the system is consistent and has exactly one solution. Visually, the two lines intersect at exactly one point on the graph.

Infinitely Many Solutions (Coincident Lines): If the ratios of all coefficients and constants are equal (i.e., $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$), the system is consistent and dependent. Visually, the two equations represent the same line, and one line lies directly on top of the other (coincident).

No Solution (Parallel Lines): If the ratios of the coefficients of $x$ and $y$ are equal, but not equal to the ratio of the constants (i.e., $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$), the system is inconsistent. Visually, the lines are parallel and maintain a constant distance from each other.

Dependency: A pair of linear equations which are equivalent (one is a non-zero multiple of the other) is called a dependent pair. A dependent pair of equations is always consistent and is visually seen as a single line on the graph.