Question

Solve the following pairs of equations by reducing them to a pair of linear equations:

$\frac{2}{\sqrt{x}}+\frac{2}{\sqrt{y}}=2$ , $\frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}=-1$

Answer 1

Given equations are $\frac{2}{\sqrt{x}}+\frac{2}{\sqrt{y}}=2$ and $\frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}=-1$

Put $\frac{1}{\sqrt{x}}=u,\frac{1}{\sqrt{y}}=v$

$\Rightarrow 2u+2v=2and4u-9v=-1$

Solving the above equations we get $u=\frac{8}{13},v=\frac{5}{13}$

Hence $\frac{1}{\sqrt{x}}=u=\frac{8}{13}\Rightarrow \sqrt{x}=\frac{13}{8}\Rightarrow x=\frac{169}{64}$ and $\frac{1}{\sqrt{y}}=v=\frac{5}{13}\Rightarrow \sqrt{y}=\frac{13}{5}\Rightarrow y=\frac{169}{25}$

Therefore $x=\frac{169}{64},y=\frac{169}{25}$