Question

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

$$\begin{gathered} \frac{1}{2x}+\frac{1}{3y}=2 \\ \frac{1}{3x}+\frac{1}{2y}=\frac{13}{6} \end{gathered}$$

Answer 1

Step 1: Reduce the pair of equations into pairs of linear equations

Let us assume that

$\frac{1}{x}=m$ and $\frac{1}{y}=n$ ,

The equations can be rewritten as

$$\begin{gathered} \frac{m}{2}+\frac{n}{3}=2 \\ \Rightarrow \frac{3m+2n}{6}=2 \\ \Rightarrow 3m+2n-12=0\dots \dots \dots \dots \dots .\left(1\right) \\ \frac{m}{3}+\frac{n}{2}=\frac{13}{6} \\ \Rightarrow \frac{2m+3n}{6}=\frac{13}{6} \\ \Rightarrow 2m+3n-13=0\dots \dots \dots \dots \dots \dots .\left(2\right) \end{gathered}$$

Step 2: Use cross multiplication method

By cross-multiplication method, we get,

$$\begin{gathered} \frac{m}{(-26-(-36)}=\frac{n}{(-24-(-39\left)\right)}=\frac{1}{(9-4)} \\ \Rightarrow \frac{m}{10}=\frac{n}{15}=\frac{1}{5} \end{gathered}$$

therefore,

$$\begin{gathered} \frac{m}{10}=\frac{1}{5} \\ \Rightarrow m=2 \\ and, \\ \frac{n}{15}=\frac{1}{5} \\ \Rightarrow n=3 \end{gathered}$$

So,

$$\begin{gathered} \frac{1}{x}=2\Rightarrow x=\frac{1}{2} \\ and, \\ \frac{1}{y}=3\Rightarrow y=\frac{1}{3} \end{gathered}$$

Hence, $\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{2}}$ and $\mathit{y}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}$ are the solution of given pair of equation.