To determine the number of solutions for the given system of equations, we can analyze the equations:
- \( -5x + y = -18 \)
- \( 10x - 2y = 36 \)
First, let's simplify the second equation. We can divide the entire equation by 2:
\[ 5x - y = 18 \]
Now, we can rewrite the system of equations:
- \( -5x + y = -18 \) (Equation 1)
- \( 5x - y = 18 \) (Equation 2)
Next, we can add both equations together to see if they simplify:
\[ (-5x + y) + (5x - y) = -18 + 18 \] This simplifies to:
\[ 0 = 0 \]
This indicates that the two equations represent the same line. Therefore, the system of equations has infinite solutions, as any point on the line is a solution to the system.
The correct response is:
infinite.