Question

Find the values of P and Q for which the following systems of equations has infinite no. of solutions
2x+3y=7;(p+q)x+(2p-q)y=21

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{equations}\mathrm{are}: \\ 2\mathrm{x}+3\mathrm{y}-7=0.....\left(1\right) \\ \left(\mathrm{p}+\mathrm{q}\right)\mathrm{x}+\left(2\mathrm{p}-\mathrm{q}\right)-21=0......\left(2\right) \\ {\mathrm{a}}_1=2;{\mathrm{b}}_1=3;{\mathrm{c}}_1=-7 \\ {\mathrm{a}}_2=\mathrm{p}+\mathrm{q};{\mathrm{b}}_2=2\mathrm{p}-\mathrm{q};{\mathrm{c}}_2=-21 \\ \mathrm{For}\mathrm{infinite}\mathrm{solutions},\mathrm{we}\mathrm{get} \\ \frac{{\mathrm{a}}_1}{{\mathrm{a}}_2}=\frac{{\mathrm{b}}_1}{{\mathrm{b}}_2}=\frac{{\mathrm{c}}_1}{{\mathrm{c}}_2} \\ \frac{2}{\mathrm{p}+\mathrm{q}}=\frac{3}{2\mathrm{p}-\mathrm{q}}=\frac{7}{21} \\ \mathrm{thus}, \\ \mathrm{p}+\mathrm{q}=6.......\left(3\right) \\ 2\mathrm{p}-\mathrm{q}=9......\left(4\right) \\ \mathrm{On}\mathrm{adding}\left(3\right)\mathrm{and}\left(4\right) \\ 3\mathrm{p}=15 \\ \mathrm{p}=5 \\ \mathrm{thus},\mathrm{q}=6-5=1 \end{gathered}$$