Question

If the pair of equations 2x + 3y – 5 = 0 and px – 6y – 8 = 0 has a unique solution for all real values of p except _______.

Answer 1

If the system of equations $a_{1}x+b_{1}y+c_1=0\mathrm{and}{a}_{2}x+b_{2}y+c_2=0$ has a unique solution, then $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$.

Since, the system of equations 2x + 3y – 5 = 0 and px – 6y – 8 = 0 has a unique solution
Therefore,
$$\begin{gathered} \frac{2}{p}\ne \frac{3}{-6} \\ \Rightarrow \frac{2}{p}\ne \frac{-1}{2} \\ \Rightarrow 4\ne -p \\ \Rightarrow p\ne -4 \end{gathered}$$

Hence, if the pair of equations 2x + 3y – 5 = 0 and px – 6y – 8 = 0 has a unique solution for all real values of p except –4.