To determine how many solutions the given system of equations has, we can solve the system either by substitution or elimination. Here, we have the system:
- \(-5x + y = -18\)
- \(10x - 2y = 36\)
Let's first manipulate the first equation to express \(y\) in terms of \(x\):
From the first equation: \[ y = 5x - 18 \]
Now, we can substitute \(y\) into the second equation:
Substituting \(y\) in the second equation gives us: \[ 10x - 2(5x - 18) = 36 \] Distributing the \(-2\): \[ 10x - 10x + 36 = 36 \] This simplifies to: \[ 36 = 36 \]
This statement is always true, indicating that the second equation does not provide any new information and is dependent on the first equation.
Since we have derived an identity from the equations, this system has infinitely many solutions (any point \((x, 5x - 18)\) will satisfy both equations).
Conclusion: The system of equations has infinitely many solutions.