Question

If the pair of equations 3x + 2ky = 2 and 2x + 5y + 1 = 0 has no solution, then k = _________.

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 no solution, then $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$.

Since, the system of equations 3x + 2ky = 2 and 2x + 5y + 1 = 0 has no solution
Therefore,
$$\begin{gathered} \frac{3}{2}=\frac{2k}{5}\ne \frac{-2}{1} \\ \Rightarrow \frac{3}{2}=\frac{2k}{5}\mathrm{and}\frac{2k}{5}\ne \frac{-2}{1} \\ \Rightarrow 15=4k\mathrm{and}2k\ne -10 \\ \Rightarrow k=\frac{15}{4}\mathrm{and}k\ne -5 \end{gathered}$$

Hence, k = $\underline{\frac{15}{4}}.$