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Addition and Subtraction of Algebraic Expression

Simplify the ...

Question

Simplify the following:
$(x - 2 y) (y - 3 x) + (x + y) (x - 3 y) - (y - 3 x) (4 x - 5 y)$

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Solution

$\begin{matrix} (x - 2 y) (y - 3 x) + (x + y) (x - 3 y) - (y - 3 x) (4 x - 5 y) = (x y - 3 x^{2} - 2 y^{2} + 6 x y) + (x^{2} - 3 x y + x y - 3 y^{2}) - (4 x y - 5 y^{2} - 12 x^{2} + 15 x y) = 7 x y - 3 x^{2} - 2 y^{2} + (x^{2} - 2 x y - 3 y^{2}) - (19 x y - 5 y^{2} - 12 x^{2}) = 7 x y - 3 x^{2} - 2 y^{2} + x^{2} - 2 x y - 3 y^{2} - 19 x y + 5 y^{2} + 12 x^{2} = (- 3 + 1 + 12) x^{2} + (7 - 2 - 19) x y + (- 2 - 3 + 5) y^{2} = 10 x^{2} - 14 x y = 2 x (5 x - 7 y) \end{matrix}$

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