Question

solve the following simultaneous equation:
m( x + y ) + n( x - y ) - ( m^2 + mn + n^2 ) = 0 ;
n( x + y ) + m( x - y ) - ( m^2 - mn + n^2 ) = 0.​

Answer 1

$\text{The given equations can be written as}$

$(m+n)x+(m-n)y=m^2+mn+n^2$...(1)

$(m+n)x-(m-n)y=m^2-mn+n^2$....(2)

$\text{Adding (1) and (2), we get}$

$2(m+n)x=2m^2+2n^2$

$\implies\boxed{\bf\,x=\frac{m^2+n^2}{m+n}}$

$\text{Put x=\frac{m^2+n^2}{m+n} in (1), we get}$

$(m+n)(\frac{m^2+n^2}{m+n})+(m-n)y=m^2+mn+n^2$

$m^2+n^2+(m-n)y=m^2+mn+n^2$

$(m-n)y=mn$

$\implies\boxed{\bf\,y=\frac{mn}{m-n}}$

$\therefore\textbf{The solution is \bf\,x=\frac{m^2+n^2}{m+n} and \bf\,y=\frac{mn}{m-n} }$

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