Question

$\frac{7}{x} + \frac{8}{y} = 2 \\ \\ and \\ \\ \frac{2}{x} + \frac{13}{y} = 22$
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Answer 1

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

The given pair of equations are

$\rm :\longmapsto\:\dfrac{7}{x} + \dfrac{8}{y} = 2 - - - (1)$

and

$\rm :\longmapsto\:\dfrac{2}{x} + \dfrac{13}{y} = 22 - - - (2)$

To solve, such pair of equations, let us assume that

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

and

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

So, above equations can be reduced to

$\rm :\longmapsto\:7u + 8v = 2 - - - (3)$

and

$\rm :\longmapsto\:2u + 13v = 22 - - - (4)$

Now, to solve these equations,

Using Cross Multiplication method, we have

$\begin{gathered}\boxed{\begin{array}{c|c|c|c} \bf 2 & \bf 3 & \bf 1& \bf 2\\ \frac{\qquad}{} & \frac{\qquad}{}\frac{\qquad}{} &\frac{\qquad}{} & \frac{\qquad}{} &\\ \sf 8 & \sf 2 & \sf 7 & \sf 8\\ \\ \sf 13 & \sf 22 & \sf 2 & \sf 13\\ \end{array}} \\ \end{gathered}$

So, we have

$\rm :\longmapsto\:\dfrac{u}{176 - 26} = \dfrac{v}{4 - 154} = \dfrac{ - 1}{91 - 16}$

$\rm :\longmapsto\:\dfrac{u}{150} = \dfrac{v}{ - 150} = \dfrac{ - 1}{75}$

$\rm :\longmapsto\:\dfrac{u}{150} = \dfrac{v}{ - 150} = \dfrac{1}{ - 75}$

On Multiply by 150, we get

$\rm :\longmapsto\:\dfrac{u}{1} = \dfrac{v}{ - 1} = - 2$

So,

$\bf\implies \:u = - 2$

and

$\bf\implies \:v = 2$

Hence,

$\bf\implies \:x = \dfrac{1}{u} = - \dfrac{1}{2}$

and

$\bf\implies \:y = \dfrac{1}{v} = \dfrac{1}{2}$

Verification :-

Consider equation (1),

$\rm :\longmapsto\:\dfrac{7}{x} + \dfrac{8}{y} = 2$

On substituting the values of x and y, we get

$\rm :\longmapsto\:7 \times ( - 2) + 8(2) = 2$

$\rm :\longmapsto\: - 14 + 16 = 2$

$\bf\implies \:2 = 2$

Hence, Verified.

Consider, equation (2),

$\rm :\longmapsto\:\dfrac{2}{x} + \dfrac{13}{y} = 22$

On substituting the values of x and y, we get

$\rm :\longmapsto\:2 \times ( - 2) + 13 \times 2 = 22$

$\rm :\longmapsto\: - 4 + 26 = 22$

$\bf\implies \:22 = 22$

Hence, Verified