To solve the system of equations:
[tex]\[
\left\{\begin{array}{l}
4 y = 8 x \\
\frac{x^2}{25} - \frac{y^2}{49} = 1
\end{array}\right.
\][/tex]
we start by isolating the variable [tex]\( y \)[/tex] in the first equation.
First, solve for [tex]\( y \)[/tex] in the equation [tex]\( 4y = 8x \)[/tex]:
[tex]\[
4y = 8x
\][/tex]
Divide both sides by 4:
[tex]\[
y = \frac{8x}{4}
\][/tex]
Simplify:
[tex]\[
y = 2x
\][/tex]
Next, we substitute [tex]\( y \)[/tex] into the second equation ([tex]\( \frac{x^2}{25} - \frac{y^2}{49} = 1 \)[/tex]) in place of [tex]\( y \)[/tex]:
[tex]\[
\frac{x^2}{25} - \frac{(2x)^2}{49} = 1
\][/tex]
Simplify inside the parentheses:
[tex]\[
\frac{x^2}{25} - \frac{4x^2}{49} = 1
\][/tex]
Thus, the resulting equation is:
[tex]\[
\frac{x^2}{25} - \frac{4x^2}{49} = 1
\][/tex]
So, the correct answer is:
C. [tex]\(\frac{x^2}{25} - \frac{4x^2}{49} = 1\)[/tex]
