To determine the number of solutions for the given system of equations, we can first rewrite the equations:
- \( -5x + y = -18 \)
- \( 10x - 2y = 36 \)
Let's simplify the second equation by dividing all terms by 2:
\[ 5x - y = 18 \]
Now we have the following system of equations:
- \( -5x + y = -18 \) (1)
- \( 5x - y = 18 \) (2)
Next, we can add both equations to eliminate \(y\):
\[ (-5x + y) + (5x - y) = -18 + 18 \]
This simplifies to:
\[ 0 = 0 \]
This statement is always true, which indicates that the two equations are dependent (i.e., they represent the same line). Therefore, there are infinitely many solutions to the system of equations.
In conclusion, the system of equations has infinitely many solutions.