Find the combined equation of the straight lines whose separate equation are $x - 2 y - 3 = 0$ and $x + y + 5 = 0.$

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Solution

$x - 2 y - 3 = 0$
$x + y + 5 = 0$
$\Rightarrow (x - 2 y - 3) (x + y + 5) = 0$
$\Rightarrow x^{2} - 2 x y - 3 x + x y - 2 y^{2} - 3 y + 5 x - 10 y - 15 = 0$
$\Rightarrow x^{2} - 2 y^{2} - x y + 2 x - 13 y - 15 = 0$

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