To determine which equation will form a system with an infinite number of solutions when graphed with the given equation \( y = 3x - 1 \), we need to identify an equation that represents the same line.
The given equation is:
\[ y = 3x - 1 \]
Next, let's analyze the options:
A. \( y + 1 = 3x \)
Rearranging this gives:
\[ y = 3x - 1 \]
This is exactly the same as the original equation, so this option will indeed create a system with an infinite number of solutions.
B. \( y = -3x + 1 \)
This line has a different slope (\(-3\)) and will not be the same line, thus it does not produce an infinite number of solutions.
C. \( y = 3x + 1 \)
This line also has the same slope (\(3\)) as the original equation but a different y-intercept (\(-1\) vs \(1\)). So it represents a different line and does not produce an infinite number of solutions.
D. \( y - 3x = -3 \)
Rearranging this gives:
\[ y = 3x - 3 \]
This line again has the same slope (\(3\)) but a different y-intercept (\(-1\) vs \(-3\)), so it represents a different line and does not have infinite solutions.
Thus, the only equation that will yield a system with an infinite number of solutions is option A:
\[ \boxed{A} \]