NCERT · Class 10 · Mathematics · Pair of Linear Equations in Two VariablesFind the value of $k$ for which the system of equations $x + 2y = 3$ and $5x + ky + 7 = 0$ has a unique solution.

The given linear equations in standard form are:

1. $1x + 2y - 3 = 0 \implies a_1 = 1, b_1 = 2, c_1 = -3$
2. $5x + ky + 7 = 0 \implies a_2 = 5, b_2 = k, c_2 = 7$ \nFor a system of linear equations to have a unique solution, the condition is: $$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$ \nSubstituting the values into the inequality: $$\frac{1}{5} \neq \frac{2}{k}$$ \nCross-multiplying gives: $$k \neq 5 \times 2 \implies k \neq 10$$ \nThus, the system of equations will have a unique solution for all real values of $k$ except $k = 10$.