To determine the number of solutions for the given system of equations, we can analyze the equations:
- \(-5x + y = -18\) (Equation 1)
- \(10x - 2y = 36\) (Equation 2)
First, let's rewrite Equation 1 in slope-intercept form (y = mx + b):
\[ y = 5x - 18 \]
Now let's rewrite Equation 2 in slope-intercept form:
\[ 10x - 2y = 36 \] \[ -2y = -10x + 36 \] \[ y = 5x - 18 \]
Now, we see that both equations simplify to the same equation:
\[ y = 5x - 18 \]
Since both equations represent the same line, they will overlap completely. This means that there are infinitely many solutions, as any point on that line will satisfy both equations.
Thus, the answer is:
infinite solutions.