MP Board · Class 10 · Mathematics · Pair of Linear Equations in Two VariablesWrite the condition for a pair of linear equations in two variables to have a unique solution. Also explain its graphical interpretation.

For a pair of linear equations $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$, the condition for having a unique solution is that the ratio of the coefficients of $x$ is not equal to the ratio of the coefficients of $y$. \nMathematically, the condition is: $$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$ \nGraphical Interpretation: When this condition holds true, the two lines represented by the linear equations intersect at exactly one point on the Cartesian coordinate plane. The coordinates $(x, y)$ of this intersecting point represent the unique solution of the system, making the pair of equations consistent.