For a system of equations to have no solution, the two lines must be parallel. This means they must have the same slope but different y-intercepts.
Given the equation \( y = 5x - 10 \), the slope is 5.
Let's analyze the provided options:
-
\( y = -5x + 10 \): This has a slope of -5. It is not parallel to \( y = 5x - 10 \).
-
\( y = 5(x + 2) \): This can be rewritten as \( y = 5x + 10 \). The slope is 5, but the y-intercept is different (10 instead of -10), making it parallel to the original equation.
-
\( y = 5(x - 2) \): This can be rewritten as \( y = 5x - 10 \). This is the same line as the original equation, so they would overlap and have infinitely many solutions.
-
\( y = -5x - 10 \): This has a slope of -5, so it is not parallel to \( y = 5x - 10 \).
The equation that will form a system with no solution is:
\( y = 5(x + 2) \) (or \( y = 5x + 10 \)).