Q $5$ $\newline$ If $4 x+4 y=9$ and $x^{2}-y^{2}=-\frac{6}{16}$ , what is the value of $2 x-2 y ?$

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Evaluate expression when two complex numbers are given

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Q. Q

5

\newline

4 x+4 y=9

and

x^{2}-y^{2}=-\frac{6}{16}

, what is the value of

2 x-2 y ?

Simplify Equation $1$ : We are given two equations: $\newline$ $1$ ) $4x + 4y = 9$ $\newline$ $2$ ) $x^2 - y^2 = -\left(\frac{6}{16}\right)$ $\newline$ First, we simplify the given equations. $\newline$ For equation $1$ , we can divide both sides by $4$ to simplify it. $\newline$ $\frac{4x + 4y}{4} = \frac{9}{4}$ $\newline$ $x + y = \frac{9}{4}$ $\newline$ $x + y = 2.25$
Simplify Equation $2$ : For equation $2$ , we simplify the fraction on the right side. $\newline$ $x^2 - y^2 = -\frac{6}{16}$ $\newline$ $x^2 - y^2 = -\frac{3}{8}$ (since $\frac{6}{16}$ simplifies to $\frac{3}{8}$ )
Find $2x - 2y$ : Now, we need to find the value of $2x - 2y$ . To do this, we can multiply the simplified form of equation $1$ by $2$ . $\newline$ $2(x + y) = 2(2.25)$ $\newline$ $2x + 2y = 4.5$
Factorize Equation $2$ : We notice that $x^2 - y^2$ is a difference of squares, which can be factored into $(x + y)(x - y)$ . So, we rewrite equation $2$ using this identity: $(x + y)(x - y) = -\frac{3}{8}$
Substitute $x + y$ : We already know the value of $x + y$ from the simplified equation $1$ , which is $2.25$ . We substitute this value into the factored form of equation $2$ . $\newline$ $2.25(x - y) = -\frac{3}{8}$
Divide by $2$ . $25$ : To find the value of $x - y$ , we divide both sides of the equation by $2.25$ .
$(x - y) = \frac{-3}{8} / 2.25$
$(x - y) = \frac{-3}{8} / \frac{9}{4}$
Simplify Multiplication: We multiply by the reciprocal of $\frac{9}{4}$ to divide by a fraction. $\newline$ $(x - y) = \left(-\frac{3}{8}\right) \times \left(\frac{4}{9}\right)$ $\newline$ $(x - y) = -3 \times 4 / 8 \times 9$
Multiply by $2$ : We simplify the multiplication. $\newline$ $(x - y) = -\frac{12}{72}$ $\newline$ $(x - y) = -\frac{1}{6}$
Multiply by $2$ : We simplify the multiplication. $\newline$ $(x - y) = -\frac{12}{72}$ $\newline$ $(x - y) = -\frac{1}{6}$ Finally, we multiply the result by $2$ to find the value of $2x - 2y$ . $\newline$ $2x - 2y = 2 \times (-\frac{1}{6})$ $\newline$ $2x - 2y = -\frac{2}{6}$ $\newline$ $2x - 2y = -\frac{1}{3}$