To determine how many solutions the given system of equations has, let's analyze the equations step-by-step.
Given system:
[tex]\[
\left\{\begin{array}{l}
5y = 15x - 40 \\
y = 3x - 8
\end{array}\right.
\][/tex]
First, let's rewrite the first equation in the slope-intercept form [tex]\( y = mx + b \)[/tex].
The first equation is:
[tex]\[
5y = 15x - 40
\][/tex]
To isolate [tex]\( y \)[/tex], divide every term by 5:
[tex]\[
y = \frac{15x - 40}{5}
\][/tex]
Simplifying the right-hand side:
[tex]\[
y = 3x - 8
\][/tex]
Now, the first equation is in the form [tex]\( y = 3x - 8 \)[/tex].
Next, consider the second equation:
[tex]\[
y = 3x - 8
\][/tex]
We see that the second equation is already in the slope-intercept form [tex]\( y = 3x - 8 \)[/tex], which is identical to the form we derived from the first equation.
Since both equations simplify to the exact same equation [tex]\( y = 3x - 8 \)[/tex], we have an identity. An identity means that every point on the line described by [tex]\( y = 3x - 8 \)[/tex] is a solution to the system of equations.
Therefore, the system has infinitely many solutions, as every value of [tex]\( x \)[/tex] and the corresponding value of [tex]\( y = 3x - 8 \)[/tex] is a solution.
Hence, the correct answer is:
(C) Infinitely many solutions
