To prove that the points $(1, -1)$, $(2, 1)$, and $(4, 5)$ are collinear, we can calculate the slope between each pair of points. If all pairs have the same slope, then the points are collinear.

Step-by-Step Solution

1. Calculate the slope between points $(1, -1)$ and $(2, 1)$: $\text{slope}_{12} = \frac{1 - (-1)}{2 - 1} = \frac{2}{1} = 2$

2. Calculate the slope between points $(2, 1)$ and $(4, 5)$: $\text{slope}_{23} = \frac{5 - 1}{4 - 2} = \frac{4}{2} = 2$

3. Calculate the slope between points $(1, -1)$ and $(4, 5)$: $\text{slope}_{13} = \frac{5 - (-1)}{4 - 1} = \frac{6}{3} = 2$

Since all three slopes are equal ($\text{slope}_{12} = \text{slope}_{23} = \text{slope}_{13} = 2$), the points $(1, -1)$, $(2, 1)$, and $(4, 5)$ are collinear.

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Related Questions

1. How do you prove collinearity if the points are given in a different format, like parametric equations?
2. Can the concept of slopes be used to determine if four or more points are collinear?
3. What happens if the slopes between pairs of points do not match?
4. How do you find the equation of a line passing through these collinear points?
5. What if the points are in 3D space? How is collinearity determined in that case?

Tip

When working with slopes to determine collinearity, ensure the points are distinct (not the same point), as identical points would automatically satisfy any slope condition.