Solve the system of equations by …

The given system of equations may be written as
x + y = a - b
So,  x + y -(a - b)=0 ......... (i)
and ax - by = a² + b²
So, ax - by -(a² + b²)=0 ........ (ii)
By cross-multiplication, using (i) and (ii) , we have
{tex}\frac { x } { - \left( a ^ { 2 } + b ^ { 2 } \right) - b ( a - b ) } = \frac { y } { - a ( a - b ) + \left( a ^ { 2 } + b ^ { 2 } \right) } = \frac { 1 } { - b - a }{/tex}
{tex}\Rightarrow \quad \frac { x } { - a ^ { 2 } - a b } = \frac { y } { a b + b ^ { 2 } } = \frac { 1 } { - b - a }{/tex}
{tex}\Rightarrow \quad \frac { x } { - a ( a + b ) } = \frac { y } { b ( a + b ) } = \frac { 1 } { - ( a + b ) }{/tex}
{tex}\Rightarrow \quad x = \frac { - a ( a + b ) } { - ( a + b ) } = a \text { and } y = \frac { b ( a + b ) } { - ( a + b ) } = - b{/tex}
Hence, x = a, y = -b is the solution of the given system of equations.