To solve the system of equations:
[tex]\[
\begin{cases}
8x - y = 8 \\
8x^2 - y = 8
\end{cases}
\][/tex]
we will work through each step methodically:
1. Solve the first equation for [tex]\(y\)[/tex]:
[tex]\[
8x - y = 8 \implies y = 8x - 8
\][/tex]
2. Substitute this expression for [tex]\(y\)[/tex] into the second equation:
[tex]\[
8x^2 - (8x - 8) = 8
\][/tex]
3. Simplify the resulting equation:
[tex]\[
8x^2 - 8x + 8 = 8
\][/tex]
[tex]\[
8x^2 - 8x + 8 - 8 = 0
\][/tex]
[tex]\[
8x^2 - 8x = 0
\][/tex]
4. Factor out the common term:
[tex]\[
8x(x - 1) = 0
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Set each factor to zero:
[tex]\[
8x = 0 \implies x = 0 \\
x - 1 = 0 \implies x = 1
\][/tex]
6. Find the corresponding [tex]\(y\)[/tex]-values for each [tex]\(x\)[/tex]:
For [tex]\(x = 0\)[/tex]:
[tex]\[
y = 8(0) - 8 = -8
\][/tex]
So one solution is [tex]\((0, -8)\)[/tex].
For [tex]\(x = 1\)[/tex]:
[tex]\[
y = 8(1) - 8 = 0
\][/tex]
So another solution is [tex]\((1, 0)\)[/tex].
Thus, the solutions to the system of equations are:
[tex]\[
\boxed{(0, -8) \text{ and } (1, 0)}
\][/tex]
Therefore, from the given options, the correct answer is:
[tex]\[
(1,0) \text{ and } (0,-8)
\][/tex]
