MP Board · Class 10 · Mathematics · Pair of Linear Equations in Two VariablesExplain the conditions for a pair of linear equations in two variables to have a unique solution, infinitely many solutions, or no solution. Also, describe their geometric representation in each case.

Consider a general pair of linear equations in two variables $x$ and $y$:

1. $a_1x + b_1y + c_1 = 0$

2. $a_2x + b_2y + c_2 = 0$ \nThe algebraic behavior and geometric nature of these equations depend on the ratios of their coefficients as follows:

3. Unique Solution (Intersecting Lines): If $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the system has a unique solution. Geometrically, the two lines represented by the equations intersect each other at exactly one point. The system is said to be consistent.

4. Infinitely Many Solutions (Coincident Lines): If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the system has infinitely many solutions. Geometrically, the two lines coincide with each other completely. The system is consistent and dependent.

5. No Solution (Parallel Lines): If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the system has no solution. Geometrically, the two lines are parallel to each other and never intersect at any point. The system is said to be inconsistent.