Question

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

$$\begin{gathered} \frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}}=2 \\ \frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}=-1 \end{gathered}$$

Answer 1

$$\begin{gathered} \frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}}=2 \\ \frac{4}{\sqrt{x}}-\frac{9}{\sqrt{y}}=-1 \end{gathered}$$

Let us substitute $\frac{1}{\sqrt{x}}=m$ and $\frac{1}{\sqrt{y}}=n$ in the given equations, we get

$$\begin{gathered} 2m+3n=2\dots \dots \dots \dots \dots \dots \dots \dots \dots ..\left(1\right) \\ 4m–9n=-1\dots \dots \dots \dots \dots \dots \dots \dots \dots \left(2\right) \end{gathered}$$

On multiplying equation $\left(1\right)$ by $3$, we get

$6m+9n=6\dots \dots \dots \dots \dots \dots \dots .\dots ..\left(3\right)$

On adding equation $\left(2\right)$ and $\left(3\right)$, we get

$$\begin{gathered} 10m=5 \\ \Rightarrow m=\frac{1}{2}\dots \dots \dots \dots \dots \dots \dots \dots \left(4\right) \end{gathered}$$

On substituting $m=\frac{1}{2}$ in equation $\left(1\right)$, we get

$$\begin{gathered} 2\times \frac{1}{2}+3n=2 \\ \Rightarrow 3n=1 \\ \Rightarrow n=\frac{1}{3} \\ m=\frac{1}{\sqrt{x}} \\ \Rightarrow \frac{1}{2}=\frac{1}{\sqrt{x}} \\ \Rightarrow x=4 \\ n=\frac{1}{\sqrt{y}} \\ \Rightarrow \frac{1}{3}=\frac{1}{\sqrt{y}} \\ \Rightarrow y=9 \end{gathered}$$

Hence, $\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4}$ and $\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{9}$ are the solution of given pair of equation.