Question

Solve for $x$ and $y$:
$mx-ny=m^2+n^2$, $x-y=2n$

• A. $x=m+n;y=m-n$
• B. $x=m-n;y=mn-n$
• C. $x=m+mn;y=m+n$
• D. $x=mn-n;y=m-n$

Answer 1

The correct option is A $x=m+n;y=m-n$

The equations given are:

$mx-ny=m^2+n^2...........\left(1\right)$

$x-y=2n..........\left(2\right)$

We pick either of the equations and write one variable in terms of the other.

Let us consider the equation 2 and write it as

$x=2n+y$

Substitute the value of $x$ in equation 1. We get

$m(2n+y)-ny=m^2+n^2\Rightarrow 2mn+my-ny=m^2+n^2\Rightarrow y(m-n)=m^2+n^2-2mn\Rightarrow y(m-n)={(m-n)}^2\Rightarrow y=m-n$

Therefore, $y=m-n$

Substituting this value of $y$ in the equation $x=2n+y$, we get

$x=2n+y=2n+m-n=m+n$

Hence, the solution is $x=m+n,y=m-n$.