Question

3/x+y+4/x-y=5,5/x+y+1/3(x-y)=2 ​

Answer 1

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

Given equations are

$\rm :\longmapsto\:\dfrac{3}{x + y} + \dfrac{4}{x - y} = 5 - - - (1)$

and

$\rm :\longmapsto\:\dfrac{5}{x + y} + \dfrac{1}{3(x - y)} = 2- - - (2)$

To solve this pair of equations, Let assume that

$\red{\rm :\longmapsto\:\dfrac{1}{x + y} = a} - - - (3)$

and

$\red{\rm :\longmapsto\:\dfrac{1}{x - y} = b} - - - (4)$

So, equation (1) and (2) can be rewritten as

$\rm :\longmapsto\:3a + 4b = 5 - - - - (5)$

$\rm :\longmapsto\:5a + \dfrac{b}{3} = 2$

$\rm :\longmapsto\:\dfrac{15a + b}{3} = 2$

$\rm :\longmapsto\:15a + b = 6 - - - (6)$

Now, multiply equation (6) by 4, we get

$\rm :\longmapsto\:60a +4 b = 24 - - - (7)$

On Subtracting equation (5) from (7), we get

$\rm :\longmapsto\:57a = 19$

$\sf\implies\boxed{\tt{ a \: = \: \frac{1}{3} \: }} - - - (8)$

On substituting the value of a in equation (5), we get

$\rm :\longmapsto\:1 + 4b = 5$

$\rm :\longmapsto\:4b = 5 - 1$

$\rm :\longmapsto\:4b = 4$

$\sf\implies\boxed{\tt{ \: b \: = \: 1 \: }} - - - - (9)$

On substituting the values of a and b in equation (3) and (4), we get

$\rm :\longmapsto\:x + y = 3 - - - - (10)$

and

$\rm :\longmapsto\:x - y = 1- - - - (11)$

On adding equation (10) and (11), we get

$\rm :\longmapsto\:2x = 4$

$\sf\implies \: x \: = \: 2$

On substituting the value of x in equation (10), we get

$\rm :\longmapsto\:2 + y = 3$

$\sf\implies \: y \: = \: 1$

So, Solution of equations is

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\bf{x \: = \: 2} \\ \\ &\bf{y \: = \: 1} \end{cases}\end{gathered}\end{gathered}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

VERIFICATION

Consider, Equation (1)

$\rm :\longmapsto\:\dfrac{3}{x + y} + \dfrac{4}{x - y} = 5$

On substituting the values of x and y, we get

$\rm :\longmapsto\:\dfrac{3}{2 + 1} + \dfrac{4}{2 - 1} = 5$

$\rm :\longmapsto\:\dfrac{3}{3} + \dfrac{4}{1} = 5$

$\rm :\longmapsto\:1 + 4= 5$

$\rm :\longmapsto\:5= 5$

Hence, Verified