To solve this system of equations:
[tex]\[
\begin{array}{l}
y = 4z \\
2z^2 - y = 0
\end{array}
\][/tex]
we will proceed step-by-step to find the solutions for \(y\) and \(z\).
1. We start with the first equation:
[tex]\[
y = 4z
\][/tex]
2. Substitute \(y = 4z\) into the second equation:
[tex]\[
2z^2 - y = 0
\][/tex]
[tex]\[
2z^2 - 4z = 0
\][/tex]
3. Simplify the second equation:
[tex]\[
2z^2 - 4z = 0
\][/tex]
4. Factor out the common term \(2z\):
[tex]\[
2z(z - 2) = 0
\][/tex]
5. Solve for \(z\):
[tex]\[
2z = 0 \quad \text{or} \quad z - 2 = 0
\][/tex]
[tex]\[
z = 0 \quad \text{or} \quad z = 2
\][/tex]
6. Substitute \(z = 0\) back into \(y = 4z\):
[tex]\[
y = 4(0) = 0
\][/tex]
Therefore, one solution is:
[tex]\[
(y, z) = (0, 0)
\][/tex]
7. Substitute \(z = 2\) back into \(y = 4z\):
[tex]\[
y = 4(2) = 8
\][/tex]
Therefore, the other solution is:
[tex]\[
(y, z) = (8, 2)
\][/tex]
So, the solution set for the system of equations is:
[tex]\[
\{(0, 0), (8, 2)\}
\][/tex]
Now, we compare the solutions to the provided options:
A. \((0.5, 0.5)\) and \((2, -3)\) - This does not match our solutions.
B. \((0, 0)\) and \((2, 8)\) - This doesn't match either since \((2, 8)\) should be \((8, 2)\).
C. \((1.5, 2)\) and \((8, 2)\) - The same mismatch issues appear here too.
D. \((-1.5, -2)\) and \((2, 8)\) - This is also incorrect in both cases.
Therefore, given the correct solution set [tex]\(\{(0, 0), (8, 2)\}\)[/tex], none of the multiple-choice options are correct.
