Solve the following pair of equations by reducing them to a pair of linear equations:
$\frac{1}{x} - \frac{4}{y} = 2 \frac{1}{x} - \frac{3}{y} = 9$

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Solution

$\frac{1}{x} - \frac{4}{y} = 2$
$\frac{1}{x} - \frac{3}{y} = 9$
Let $\frac{1}{x} = u, \frac{1}{y} = v$
So, $u - 4 v = 2$ ...(i)
$u - 3 v = 9$ ...(ii)
On solving eq. (i) and eq. (ii)
$u - 4 v = 2$
$u - 3 v = 9$
$- -$
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$- v = - 7$
$v = 7$
Putting the value of v in eq. (i)
$u - 4 v = 2$
$u - 4 \times 7 = 2$
$u - 28 = 2$
$u = 2 + 28$
$u = 30$
So $v = 7 \Rightarrow \frac{1}{y} = 7, y = \frac{1}{7}$
$u = 30 \Rightarrow \frac{1}{x} = 30, x = \frac{1}{30}$
Hence, $x = \frac{1}{30}$ and $y = \frac{1}{7}$

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

\frac{2}{\sqrt{x}} + \frac{3}{\sqrt{y}} = 2, \frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}} = - 1

\frac{2}{\sqrt{x}} + \frac{2}{\sqrt{y}} = 2

\frac{4}{\sqrt{x}} - \frac{9}{\sqrt{y}} = - 1

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