To determine the number of solutions for the given system of equations:
[tex]\[
\begin{cases}
y = \frac{5}{2} x + 2 \\
2y = 5x + 4
\end{cases}
\][/tex]
we can start by analyzing whether the two equations are consistent. If they intersect at a single point, there will be one unique solution. If they are the same line (i.e., coincident), they will have infinitely many solutions. If they are parallel and distinct, there will be no solutions.
Let's start by seeing if the two equations can provide us with one common point.
First, we can rewrite the second equation in a form that makes it easier to compare with the first. Recall that we have:
[tex]\[
2y = 5x + 4
\][/tex]
Divide every term by 2 to write it in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = \frac{5}{2} x + 2
\][/tex]
Now, we see that the first equation is:
[tex]\[
y = \frac{5}{2} x + 2
\][/tex]
And the rewritten second equation is:
[tex]\[
y = \frac{5}{2} x + 2
\][/tex]
Both equations are actually the same. This indicates that every point (x, y) that satisfies one equation will also satisfy the other, meaning the two equations represent the same line.
Hence, because they represent the same line, the system has infinitely many solutions.
Thus, the correct answer is:
C. Infinitely many
