To solve the given system of equations:
[tex]\[
\begin{aligned}
2 x + 8 y &= 5 \\
24 x - 4 y &= -15
\end{aligned}
\][/tex]
we will use the method of substitution or elimination. We can choose either method, but here we will outline the steps assuming we use the elimination method to make it clear.
1. First, let’s rewrite the equations to make it easier to eliminate one of the variables:
[tex]\[
\begin{aligned}
2x + 8y &= 5 \quad \text{(Equation 1)} \\
24x - 4y &= -15 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
2. To eliminate one variable, we can multiply Equation 1 by a factor so that the coefficients of one of the variables match up when we add or subtract the equations. Here, let's try to eliminate [tex]\( y \)[/tex].
Multiply Equation 1 by [tex]\(2\)[/tex]:
[tex]\[
\begin{aligned}
4x + 16y &= 10 \quad \text{(Equation 3)}
\end{aligned}
\][/tex]
3. Now we have:
[tex]\[
\begin{aligned}
4x + 16y &= 10 \quad \text{(Equation 3 from above)} \\
24x - 4y &= -15 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
4. Next, we add Equation 2 and Equation 3 to eliminate [tex]\( y \)[/tex]:
Multiply Equation 2 by [tex]\(4\)[/tex] to balance the coefficients of [tex]\( y \)[/tex]:
[tex]\[
\begin{aligned}
96x - 16y &= -60 \quad \text{(Equation 4)}
\end{aligned}
\][/tex]
5. Now add Equation 3 and Equation 4:
[tex]\[
\begin{aligned}
(4x + 16y) + (96x - 16y) &= 10 + (-60) \\
100x &= -50 \\
x &= \frac{-50}{100} \\
x &= -\frac{1}{2}
\end{aligned}
\][/tex]
6. Substitute [tex]\( x = -\frac{1}{2} \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. We use Equation 1:
[tex]\[
\begin{aligned}
2\left(-\frac{1}{2}\right) + 8y &= 5 \\
-1 + 8y &= 5 \\
8y &= 6 \\
y &= \frac{6}{8} \\
y &= \frac{3}{4}
\end{aligned}
\][/tex]
The solution to the system is [tex]\( \left( -\frac{1}{2}, \frac{3}{4} \right) \)[/tex].
Therefore, the answers to fill in the boxes are:
[tex]\[
\left( \boxed{-\frac{1}{2}}, \boxed{\frac{3}{4}} \right)
\][/tex]
