Question

solve for X and y by using method of substitution. xy/x+y=6/5, xy/y-x=6​

Answer 1

$\large\underline{\sf{Solution-}}$

Given pair of equations are .

$\rm :\longmapsto\:\dfrac{xy}{x + y} = \dfrac{6}{5} - - - (1)$

and

$\rm :\longmapsto\:\dfrac{xy}{y - x} = 6 - - - (2)$

Now,

Equation (1) can be rewritten as

$\rm :\longmapsto\:\dfrac{x + y}{xy} = \dfrac{5}{6}$

$\rm :\longmapsto\:\dfrac{x}{xy} + \dfrac{y}{xy} = \dfrac{5}{6}$

$\rm :\longmapsto\:\dfrac{1}{y} + \dfrac{1}{x} = \dfrac{5}{6}$

$\rm :\longmapsto\:\dfrac{1}{x} = \dfrac{5}{6} - \dfrac{1}{y} - - - (3)$

Now,

Equation (2) can be rewritten as

$\rm :\longmapsto\:\dfrac{y - x}{xy} = \dfrac{1}{6}$

$\rm :\longmapsto\:\dfrac{y}{xy} - \dfrac{x}{xy} = \dfrac{1}{6}$

$\rm :\longmapsto\:\dfrac{1}{x} - \dfrac{1}{y} = \dfrac{1}{6}$

On substituting the value from equation (3), we get

$\rm :\longmapsto\:\dfrac{5}{6} - \dfrac{1}{y} - \dfrac{1}{y} = \dfrac{1}{6}$

$\rm :\longmapsto\:\dfrac{5}{6} - \dfrac{2}{y} = \dfrac{1}{6}$

$\rm :\longmapsto\:\dfrac{2}{y} = \dfrac{5}{6} - \dfrac{1}{6}$

$\rm :\longmapsto\:\dfrac{2}{y} = \dfrac{5 - 1}{6}$

$\rm :\longmapsto\:\dfrac{2}{y} = \dfrac{4}{6}$

$\rm :\longmapsto\:\dfrac{2}{y} = \dfrac{2}{3}$

$\rm :\longmapsto\:\dfrac{1}{y} = \dfrac{1}{3}$

$\bf\implies \:y = 3$

On substituting y = 3 in equation (3), we get

$\rm :\longmapsto\:\dfrac{1}{x} = \dfrac{5}{6} - \dfrac{1}{3}$

$\rm :\longmapsto\:\dfrac{1}{x} = \dfrac{5}{6} - \dfrac{2}{6}$

$\rm :\longmapsto\:\dfrac{1}{x} = \dfrac{5 - 2}{6}$

$\rm :\longmapsto\:\dfrac{1}{x} = \dfrac{3}{6}$

$\rm :\longmapsto\:\dfrac{1}{x} = \dfrac{1}{2}$

$\bf\implies \:x = 2$

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:Hence-\begin{cases} &\bf{x = 2} \\ &\bf{y = 3} \end{cases}\end{gathered}\end{gathered}$

Verification :-

Consider equation (1),

$\rm :\longmapsto\:\dfrac{xy}{x + y} = \dfrac{6}{5}$

On substituting x = 2 and y = 3, we get

$\rm :\longmapsto\:\dfrac{(2)(3)}{2 + 3} = \dfrac{6}{5}$

$\rm :\longmapsto\:\dfrac{6}{5} = \dfrac{6}{5}$

Hence, Verified

Consider equation (2),

$\rm :\longmapsto\:\dfrac{xy}{y - x} = 6$

On substituting x = 2 and y = 3, we get

$\rm :\longmapsto\:\dfrac{(2)(3)}{3 - 2} = 6$

$\rm :\longmapsto\:\dfrac{6}{1} = 6$

$\rm :\longmapsto\:6 = 6$

Hence, Verified