Solve the following system of equation: x+y=a-b and ax-by=a^2+b^2

1.	Solve the following system of equation: x+y=a-b and ax-by=a^2+b^2
Answer» The given system of equations may be written asx + y = a - bSo, x + y -(a - b)=0 ......... (i)and ax - by = a2 + b2So, ax\xa0- 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.

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