Question

Determine the value of α, for which the following pair of linear equations has infinitely many solutions 2x - y = 5 and 4x +αy = 10

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{system}\mathrm{of}\mathrm{equations}\mathrm{is}: \\ 2\mathrm{x}-\mathrm{y}-5=0....\left(1\right) \\ 4\mathrm{x}+\mathrm{\alpha y}-10=0.....\left(2\right) \\ \mathrm{Comparing}\mathrm{the}\mathrm{above}\mathrm{equations}\mathrm{with}{\mathrm{a}}_1\mathrm{x}+{\mathrm{b}}_1\mathrm{y}+{\mathrm{c}}_1=0\mathrm{and}{\mathrm{a}}_2\mathrm{x}+{\mathrm{b}}_2\mathrm{y}+{\mathrm{c}}_2=0,\mathrm{we}\mathrm{get} \\ {\mathrm{a}}_1=2;{\mathrm{b}}_1=-1;{\mathrm{c}}_1=-5 \\ {\mathrm{a}}_2=4;{\mathrm{b}}_2=\mathrm{\alpha };{\mathrm{c}}_2=-10 \\ \mathrm{Now},\mathrm{for}\mathrm{infinitely}\mathrm{many}\mathrm{solutions}, \\ \frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}=\frac{{\mathrm{c}}_1}{{\mathrm{c}}_2} \\ \Rightarrow \frac{2}{4}=\frac{-1}{\mathrm{\alpha }}=\frac{-5}{-10} \\ \Rightarrow \frac{1}{2}=-\frac{1}{\mathrm{\alpha }}=\frac{1}{2} \\ \Rightarrow \mathrm{\alpha }=-2 \end{gathered}$$