To determine how many solutions the system of equations has, we can analyze the two equations:
- \(-5x + y = -18\)
- \(10x - 2y = 36\)
First, we can rearrange both equations to the slope-intercept form \(y = mx + b\), which makes it easier to compare their slopes.
Rearranging the first equation: \[ y = 5x - 18 \]
Rearranging the second equation: \[ 10x - 2y = 36 \] \[ -2y = -10x + 36 \] \[ y = 5x - 18 \]
Now we have both equations in the form of \(y = 5x - 18\).
Since both equations are identical, they represent the same line. This means that there are infinitely many solutions to this system of equations, as any point on the line \(y = 5x - 18\) is a solution.
Therefore, the answer is infinite solutions.