Solve : (a-b) x+(a+b) y=a^2-2 a b-b^2 and (a+b)(x+y)=a^2+b^2 - Vidyadip

Question

Solve : $(a-b) x+(a+b) y=a^2-2 a b-b^2$ and $(a+b)(x+y)=a^2+b^2$

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Answer

$(a-b) x+(a+b) y=a^2-2 a b-b^2$...(i)
$(a+b)(x+y) \quad=a^2+b^2$
$\therefore \quad(a+b) x+(a+b) y=a^2+b^2$...(ii)
Subtracting (ii) from (i),

$\therefore \quad(a-b-a-b) x=-2 b(a+b)$
$\therefore \quad-2 b x=-2 b(a+b)$
$\therefore \quad x=\frac{-2 b(a+b)}{-2 b}$
$\therefore \quad x=(a+b)$
Substituting value of x in (i), we get,
$(a-b)(a+b)+(a+b) y=a^2-2 a b-b^2$
$\therefore \quad a^2-b^2+(a+b) y=a^2-2 a b-b^2$
$\therefore \quad(a+b) y=-2 a b$
$\therefore \quad y=\frac{-2 a b}{a+b}$
$\therefore$$x=(a+b)$ and $y=\frac{-2 a b}{a+b}$ is the solution of the given equations.

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