If $3 x - 2 y = 5$ and $x y = 4$ then find the value of $9 x^{2} + 4 y^{2}$

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Solution

Given:-
$3 x - 2 y = 5 \dots (i)$
$x y = 4 \dots (i i)$

By squaring equation $(i),$
$\begin{matrix} {(3 x - 2 y)}^{2} & = 5^{2} {(3 x)}^{2} - 2 (3 x) (2 y) + {(2 y)}^{2} & = 25 9 x^{2} + 4 y^{2} - 12 x y & = 25 9 x^{2} + 4 y^{2} & = 25 + 12 x y \dots (i i i) \end{matrix}$

Now, by using equation $(i i)$ and $(i i i),$
$\begin{matrix} 9 x^{2} + 4 y^{2} & = 25 + 12 x y = 25 + 12 (4) = 25 + 48 = 73 \end{matrix}$

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(i) $x^{2} + y^{2} + z^{2} - x y + x + y z by x + y - z$

(ii) $x^{2} + 4 y^{2} + z^{2} + 2 x y + x z - 2 y z by x - 2 y - z$

(iii) $x^{2} + 4 y^{2} + 2 x y - 3 x + 6 y + 9 by x - 2 y + 3$

(iv) $9 x^{2} + 25 y^{2} + 15 x y + 2 x - 20 y + 16 by 3 x - 5 y + 4$

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