Question

Solve x and y , m/x-n/y=a,qx-py=0

Answer 1

$y=\frac{1}{a}[\frac{mq}{p}-n]$ and $x=\frac{1}{aq}(mq-np)$

Explanation:

Given equations are

$\frac{m}{x}-\frac{n}{y}=a$          ..... (1)

$qx-py=0$         ... (2)

From equation (2) we get

$qx=py$

$x=\dfrac{py}{q}$                .... (3)

Substitute $x=\frac{py}{q}$ in equation (1).

$\frac{m}{\frac{py}{q}}-\frac{n}{y}=a$

$\frac{mq}{py}-\frac{n}{y}=a$

$\frac{1}{y}[\frac{mq}{p}-n]=a$

$\frac{1}{a}[\frac{mq}{p}-n]=y$

The value of y is $\frac{1}{a}[\frac{mq}{p}-n]$.

Substitute $y=\frac{1}{a}[\frac{mq}{p}-n]$ in equation (3).

$x=\dfrac{p}{q}(\frac{1}{a}[\frac{mq}{p}-n])$

$x=\dfrac{p}{q}(\frac{1}{a}(\frac{mq-np}{p}))$

$x=\frac{1}{aq}(mq-np)$

Therefore, the value of x is $x=\frac{1}{aq}(mq-np)$.

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Answer 2

The value of x is $\dfrac{1}{ap}(mq-np)$ and the value of y is $\dfrac{1}{a}(\dfrac{qm}{p}-n)$.

Explanation:

The given system of equations :

$\dfrac{m}{x}-\dfrac{n}{y}=a---------(1)\\\\ qx-py=0------------(2)$

From (2) , we have

$qx=py$

i.e. $x=\dfrac{p}{q}y ----------(3)$  [Divide both sides by q]      

Substitute the value of x in (1) , we get

$\dfrac{m}{\dfrac{p}{q}y}-\dfrac{n}{y}=a\\\\\\\Rightarrow\ \dfrac{qm}{py}-\dfrac{n}{y}=a\\\\\Rightarrow\ \dfrac{qm-pn}{py}=a\\\\\Rightarrow\ qm-pn=apy\\\\\Rightarrow\ \dfrac{qm-pn}{ap}=y\\\\ \Rightarrowy=\dfrac{qm}{ap}-\dfrac{n}{a}=\dfrac{1}{a}(\dfrac{qm}{p}-n)$

Put value of y in (3) ,  $x=\dfrac{p}{q}(\dfrac{1}{a}(\dfrac{qm}{p}-n))$

$x=\dfrac{1}{ap}(mq-np)$

Therefore , the value of x is $\dfrac{1}{ap}(mq-np)$ and the value of y is $\dfrac{1}{a}(\dfrac{qm}{p}-n)$.

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