Question

(v) 64p − 45q = 289; 45p − 64q = 365

Answer 1

$$\begin{gathered} 64p-45q=289...\left(\mathrm{i}\right) \\ 45p-64q=365...\left(\mathrm{ii}\right) \\ \mathrm{Adding}\mathrm{eq}.\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right),\text{we get:} \\ 64p-45q=289 \\ \underline{45p-64q=365} \\ 109p-109q=654 \\ \therefore 109\left(p-q\right)=654 \\ \Rightarrow \left(p-q\right)=6 \\ \mathrm{Subtracting}\mathrm{eq}\left(\mathrm{ii}\right)\mathrm{from}\left(\mathrm{i}\right),\text{we get:} \\ 64p-45q=289 \\ 45p-64q=365 \\ \underline{\underline{-+-}} \\ 19p+19q=-76 \\ \therefore 19\left(p+q\right)=-76 \\ \therefore \left(p+q\right)=-4...\left(\mathrm{iii}\right) \\ \&\left(p-q\right)=6...\left(\mathrm{iv}\right) \\ \mathrm{Adding}\mathrm{eq}\left(\mathrm{iii}\right)\mathrm{and}\left(\mathrm{iv}\right),\text{we get:} \\ p+q=-4 \\ +\underline{p-q=6} \\ 2p=2 \\ \therefore p=1 \\ \mathrm{Substituting}p=1\mathrm{in}\mathrm{eq}\left(\mathrm{iii}\right),\text{we get:} \\ p+q=-4 \\ \Rightarrow 1+q=-4 \\ \therefore q=-5 \\ \mathrm{Hence},p=1\mathrm{and}q=-5\mathrm{is}\mathrm{the}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{equations}. \end{gathered}$$