Question

2/x-3/y=5;8/x+5/y=77 give me the answer please yaar my is not coming properly carrect​

Answer 1

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

Given pair of linear equations are

$\rm :\longmapsto\:\dfrac{2}{x} - \dfrac{3}{y} = 5$

and

$\rm :\longmapsto\:\dfrac{8}{x} + \dfrac{5}{y} = 77$

Let us assume that

$\rm :\longmapsto\:\dfrac{1}{x} = u - - - (1)$

and

$\rm :\longmapsto\:\dfrac{1}{y} = v - - - (2)$

So, above equations can be rewritten as

$\rm :\longmapsto\:2u - 3v = 5 - - - - (3)$

and

$\rm :\longmapsto\:8u + 5v = 77- - - - (4)$

On multiply equation (3) by 4, we get

$\rm :\longmapsto\:8u - 12v = 20 - - - - (5)$

On Subtracting equation (5) from equation (3), we get

$\rm :\longmapsto\:17v = 57$

$\bf\implies \:v = \dfrac{57}{17}$

On substituting the value of v in equation (3), we get

$\rm :\longmapsto\:2u - 3 \times \dfrac{57}{17} = 5$

$\rm :\longmapsto\:2u - \dfrac{171}{17} = 5$

$\rm :\longmapsto\:2u = 5 + \dfrac{171}{17}$

$\rm :\longmapsto\:2u = \dfrac{85 + 171}{17}$

$\rm :\longmapsto\:2u = \dfrac{256}{17}$

$\rm :\longmapsto\:u = \dfrac{128}{17}$

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf\: So-\begin{cases} &\sf{u = \dfrac{128}{17} } \\ \\ &\sf{v = \dfrac{57}{17} } \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf\: So-\begin{cases} &\sf{x = \dfrac{17}{128} } \\ \\ &\sf{y = \dfrac{17}{57} } \end{cases}\end{gathered}\end{gathered}$

Verification :-

Consider,

$\rm :\longmapsto\:\dfrac{2}{x} - \dfrac{3}{y} = 5$

On substituting the values of x and y, we get

$\rm :\longmapsto\:\dfrac{2 \times 128}{17} - \dfrac{3 \times 57}{17} = 5$

$\rm :\longmapsto\:\dfrac{256}{17} - \dfrac{171}{17} = 5$

$\rm :\longmapsto\:\dfrac{256 - 171}{17} = 5$

$\rm :\longmapsto\:\dfrac{85}{17} = 5$

$\bf\implies \:5 = 5$

Hence, Verified