To determine the number of solutions in the system of equations given:
- \( -5x + y = -18 \) (Equation 1)
- \( 10x - 2y = 36 \) (Equation 2)
We can solve these equations to either find a unique solution, find no solutions, or find infinitely many solutions.
First, let's rearrange Equation 1 to solve for \( y \):
\[ y = 5x - 18 \]
Now we can substitute this expression for \( y \) into Equation 2:
\[ 10x - 2(5x - 18) = 36 \]
Simplifying this:
\[ 10x - 10x + 36 = 36 \]
This simplifies to:
\[ 36 = 36 \]
This is a true statement, meaning that the two equations represent the same line. Therefore, there are infinitely many solutions to the system.
The answer is: infinite.