Question

For what value of k will the equation x +5y-7=0 and 4x +20y +k=0 represent coincident lines?

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{system}\mathrm{of}\mathrm{equations}\mathrm{is}: \\ \mathrm{x}+5\mathrm{y}-7=0 \\ 4\mathrm{x}+20\mathrm{y}+\mathrm{k}=0 \\ \mathrm{Comparing}\mathrm{the}\mathrm{above}\mathrm{equations}\mathrm{with}{\mathrm{a}}_1\mathrm{x}+{\mathrm{b}}_1\mathrm{y}+{\mathrm{c}}_1=0\mathrm{and}{\mathrm{a}}_2\mathrm{x}+{\mathrm{b}}_2\mathrm{y}+{\mathrm{c}}_2=0,\mathrm{we}\mathrm{get} \\ {\mathrm{a}}_1=1;{\mathrm{b}}_1=5;{\mathrm{c}}_1=-7 \\ {\mathrm{a}}_2=4;{\mathrm{b}}_2=20;{\mathrm{c}}_2=\mathrm{k} \\ \mathrm{For}\mathrm{coincident}\mathrm{lines}: \\ \frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}=\frac{{\mathrm{c}}_1}{{\mathrm{c}}_2} \\ \Rightarrow \frac{1}{4}=\frac{5}{20}=\frac{-7}{\mathrm{k}} \\ \Rightarrow \frac{1}{4}=\frac{1}{4}=-\frac{7}{\mathrm{k}} \\ \Rightarrow \boxed{\mathbf{k}\mathbf{}\mathbf{=}\mathbf{}\mathbf{-}\mathbf{28}} \end{gathered}$$