To determine the number of solutions for the given system of equations, we can first rearrange the equations and then analyze their slopes and intercepts.
The system is:
- \(-5x + y = -18\)
- \(10x - 2y = 36\)
Let's rewrite them in slope-intercept form \(y = mx + b\).
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From \(-5x + y = -18\): \[ y = 5x - 18 \]
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From \(10x - 2y = 36\), we can rearrange it: \[ -2y = -10x + 36 \implies y = 5x - 18 \]
Now, we see that both equations simplify to the same line: \(y = 5x - 18\).
Since both equations represent the same line, they intersect at infinitely many points. Therefore, the system of equations has infinite solutions.
So the answer is: infinite.