If $2 x^{2} + y^{2} - 3 x y - 3 x + 3 y = 0,$ then find the value of $2 x - y$ . (Given: $x \neq y)$

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Solution

The correct option is B 3
Given:
$2 x^{2} + y^{2} - 3 x y - 3 x + 3 y = 0 - - (1)$

Let us try to factorize the expression on the left.

$2 x^{2} + y^{2} - 3 x y - 3 x + 3 y$
$= 2 x^{2} - x y - 3 x + y^{2} - 2 x y + 3 y$
$= x (2 x - y - 3) - y (- y + 2 x - 3)$
$= x (2 x - y - 3) - y (2 x - y - 3)$
$= (x - y) (2 x - y - 3)$

$∴ (x - y) (2 x - y - 3) = 0 (from (1))$

It is given that $x \neq y .$
$⟹ x - y \neq 0$

$⟹ 2 x - y - 3 = 0$
I.e., $2 x - y = 3$

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