Question

If the two lines given by 3x + 2ky = 2 and 2x + 5y + 1 = 0 are parallel, then find the value of k.

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{system}\mathrm{of}\mathrm{equations}\mathrm{is}: \\ 3\mathrm{x}+2\mathrm{ky}-2=0.....\left(1\right) \\ 2\mathrm{x}+5\mathrm{y}+1=0......\left(2\right) \\ \mathrm{Comparing}\mathrm{the}\mathrm{above}\mathrm{equations}\mathrm{with}{\mathrm{a}}_1\mathrm{x}+{\mathrm{b}}_1\mathrm{y}+{\mathrm{c}}_1=0\mathrm{and}{\mathrm{a}}_2\mathrm{x}+{\mathrm{b}}_2\mathrm{y}+{\mathrm{c}}_2=0,\mathrm{we}\mathrm{get} \\ {\mathrm{a}}_1=3;{\mathrm{b}}_1=2\mathrm{k};{\mathrm{c}}_1=-2 \\ {\mathrm{a}}_2=2;{\mathrm{b}}_2=5;{\mathrm{c}}_2=1 \\ \mathrm{For}\mathrm{the}\mathrm{lines}\mathrm{to}\mathrm{be}{\mathrm{parallel}}_{,} \\ \frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}\ne \frac{{\mathrm{c}}_1}{{\mathrm{c}}_2} \\ \Rightarrow \frac{3}{2}=\frac{2\mathrm{k}}{5}\ne \frac{-2}{1} \\ \Rightarrow \frac{3}{2}=\frac{2\mathrm{k}}{5}\mathrm{and}\frac{2\mathrm{k}}{5}\ne \frac{-2}{1} \\ \Rightarrow \mathrm{k}=\frac{15}{4}\mathrm{and}\mathrm{k}\ne -5 \\ \mathrm{So},\mathrm{the}\mathrm{lines}\mathrm{represented}\mathrm{by}\mathrm{the}\mathrm{given}\mathrm{equations}\mathrm{will}\mathrm{be}\mathrm{parallel}\mathrm{for}\mathrm{k}=\frac{15}{4} \end{gathered}$$