Question

What is the solution for the following pair of linear equations?

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

$\frac{1}{x}+\frac{1}{2y}=8$

(where $x\ne 0,y\ne 0)$

• A. $x=\frac{1}{6},y=\frac{1}{4}$
• B. $x=\frac{1}{7},y=\frac{1}{2}$
• C. $x=\frac{1}{8},y=\frac{1}{3}$
• D. $x=\frac{3}{4},y=\frac{5}{4}$

Answer 1

The correct option is A $x=\frac{1}{6},y=\frac{1}{4}$

We have
$\frac{1}{2x}-\frac{1}{y}=-1\dots \left(1\right)$
$\frac{1}{x}+\frac{1}{2y}=8...\left(2\right)$

From equation (1),
$\frac{1}{2x}=-1+\frac{1}{y}$
$\Rightarrow \frac{1}{2x}=\frac{(-y+1)}{y}$
$\Rightarrow 2x=\frac{y}{(-y+1)}$
$\Rightarrow x=\frac{y}{(-2y+2)}$

$Puttingx=\frac{y}{(-2y+2)}$ in equation (2),
$\frac{(-2y+2)}{y}+\frac{1}{2y}=8$
$\Rightarrow \frac{(-4y+4+1)}{2y}=8$
$\Rightarrow -4y+4+1=16y$
$\Rightarrow -20y=-5$
$\Rightarrow y=\frac{1}{4}$

$Puttingy=\frac{1}{4}$ in equation (1),
$\Rightarrow \frac{1}{2x}-4=-1$
$\Rightarrow \frac{1}{2x}=3$
$\Rightarrow x=\frac{1}{6}$