Question 7 (iv)
Solve the following pair of linear equations.
$(a - b) x + (a + b) y = a^{2} - 2 a b - b^{2}$
$(a + b) (x + y) = a^{2} + b^{2}$

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Solution

$(a - b) x + (a + b) y = a^{2} - 2 a b - b^{2}$ ....(1)
$(a + b) (x + y) = a^{2} + b^{2}$
$(a + b) x + (a + b) y = a^{2} + b^{2}$ .....(2)
Subtracting equation (2) from (1), we obtain
$(a - b) x - (a + b) x = (a^{2} - 2 a b - b^{2}) - (a^{2} + b^{2})$
$(a - b - a - b) x = - 2 a b - 2 b^{2}$
$- 2 b x = - 2 b (a + b)$
$x = a + b$
Using equation (1), we obtain
$(a - b) (a + b) + (a + b) y = a^{2} - 2 a b - b^{2}$
$a^{2} - b^{2} + (a + b) y = a^{2} - 2 a b - b^{2}$
$(a + b) y = - 2 a b$
$y = \frac{- 2 a b}{a + b}$

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