Question

Based on equations reducible to linear equations, solve for $x$ and $y$:

$\frac{x-y}{xy}=9;\frac{x+y}{xy}=5$

• A. $x=-\frac{1}{2},y=\frac{1}{7}$
• B. $x=-\frac{1}{5},y=\frac{1}{2}$
• C. $x=-\frac{1}{5},y=\frac{1}{7}$
• D. None of these

Answer 1

Answer: (A)

$x=-\frac{1}{2},y=\frac{1}{7}$

The given equations are
$\Rightarrow \frac{x-y}{xy}=9\Rightarrow x-y=9xy....eq\left(1\right)$
$\Rightarrow \frac{x+y}{xy}=5\Rightarrow x+y=5xy....eq\left(2\right)$
Divide both eq by $xy$
$\Rightarrow \frac{1}{y}-\frac{1}{x}=\mathrm{9....}eq\left(3\right)$
$\Rightarrow \frac{1}{y}+\frac{1}{x}=\mathrm{5....}eq\left(4\right)$
Put $\frac{1}{x}=u$ and $\frac{1}{y}=v$ in eq $\left(3\right)$ and $\left(4\right)$
$\Rightarrow v-u=9....eq\left(5\right)$
$\Rightarrow v+u=5....eq\left(6\right)$
Subtract eq $\left(5\right)$ and eq$\left(6\right)$
$\Rightarrow (v-u=9)-(v+u=5)$
$\Rightarrow 2v=-14\Rightarrow v=7$
Put $v=7$ in eq$\left(5\right)$
$\Rightarrow 7+u=5\Rightarrow u=-2$
Hence, $\frac{1}{x}=u=-2\Rightarrow x=-\frac{1}{2}$
$\frac{1}{y}=v=7\Rightarrow y=\frac{1}{7}$