Question

Solve for x and y:

$\frac{5}{x}-\frac{3}{y}=1,\frac{3}{2x}+\frac{2}{3y}=5(x\ne 0,y\ne 0).$

Answer 1

$\frac{5}{x}$ - $\frac{3}{y}$ = 1........(1)
and $\frac{3}{2x}$ + $\frac{2}{3y}$ = 5......(2)


From equation (1) transposing we have
$\to$$\frac{5}{x}$-1= $\frac{3}{y}$
$\to$$\frac{5-x}{x}$ = $\frac{3}{y}$

Applying invertendo we have

$\frac{x}{5-x}$ = $\frac{y}{3}$

$\to$ $\frac{3x}{5-x}$ = y ....( value of y)

Put the value of y in equation (2) which is

$\frac{3}{2x}$ + $\frac{2}{3y}$ = 5 we have

$\to$ $\frac{3}{2x}$ + $\frac{2}{3\frac{3x}{5-x}}$ = 5

$\to$ $\frac{3}{2x}$ + $\frac{2(5-x)}{9x}$ = 5

Since the Common Denominator= 18 we have

27 +4 (5-x) = $18x\times 5$

$\to$ 27 + 20 -4x = 90 x

$\to$ 47 = 90x + 4x

$\to$ 47= 94 x

$\to$ $\frac{47}{94}$ = x

$\to$ $\frac{1}{2}$ = x

Put the value of x in equation (1) $\frac{5}{x}$ - $\frac{3}{y}$ = 1

we have
$\frac{5}{x}$ - $\frac{3}{y}$ = 1 substituting for $\frac{1}{2}$ = x

$\to$ $\frac{5}{\frac{1}{2}}$ - $\frac{3}{y}$ = 1

$\to$ $\frac{5x2}{1}$ - $\frac{3}{y}$ = 1

$\to$ $\frac{10}{1}$ - $\frac{3}{y}$ = 1

$\to$ 10-1 = $\frac{3}{y}$
$\to$ 9 = $\frac{3}{y}$

$\to$ y= $\frac{3}{9}$

or y= $\frac{1}{3}$

So x= $\frac{1}{2}$ and y= $\frac{1}{3}$