To find two equations that form a system with no solutions (i.e., they are parallel lines), we need to identify two equations with the same slope but different y-intercepts.
Let's analyze the options:
- \(y = 5x - 4\): Slope = 5, y-intercept = -4
- \(y = -2x + 3\): Slope = -2, y-intercept = 3
- \(y = 37 + 1\): This simplifies to \(y = 38\) (a horizontal line). Slope = 0, y-intercept = 38
- \(2y = 6x + 2\): Dividing everything by 2 gives \(y = 3x + 1\). Slope = 3, y-intercept = 1
- \(y - 5x = 2\): Rearranging gives \(y = 5x + 2\). Slope = 5, y-intercept = 2
- \(y - 4x + 1 = 0\): Rearranging gives \(y = 4x - 1\). Slope = 4, y-intercept = -1
Now, let's summarize the slopes and y-intercepts:
- Option 1: Slope = 5, y-intercept = -4
- Option 2: Slope = -2, y-intercept = 3
- Option 3: Slope = 0, y-intercept = 38
- Option 4: Slope = 3, y-intercept = 1
- Option 5: Slope = 5, y-intercept = 2
- Option 6: Slope = 4, y-intercept = -1
The equations with the same slope (parallel lines) are:
- Option 1: \(y = 5x - 4\) and Option 5: \(y = 5x + 2\)
Since both have a slope of 5 but different y-intercepts (-4 and 2), they are parallel and will have no solutions.
Therefore, the answer is: Option 1 and Option 5 form a system with no solutions.