Question

$$\begin{gathered} mx-ny=m^2+n^2, \\ x+y=2m. \end{gathered}$$

Answer 1

The given equations are:
mx − ny = m² + n²........(i)
x + y = 2m ........(ii)

On multiplying (ii) by n, we get:
nx + ny = 2mn .......(iii)

On adding (i) and (iii), we get:
mx + nx = m² + n² + 2mn
⇒ (m + n)x = (m + n)²
⇒ $x=\frac{{\left(m+n\right)}^2}{\left(m+n\right)}=m+n$

On substituting x = m + n in (i), we get:
m(m + n) − ny = m² + n²
⇒ m² + mn − ny = m² + n²
⇒ −ny = n² − mn
⇒ $y=\frac{n\left(n-m\right)}{-n}=m-n$

Hence, the solution is x = (m + n) and y = (m − n).