1. **State the problem:** Find the slope of the line passing through the points $(5,-1)$ and $(-1,5)$.\n\n2. **Formula for slope:** The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:\n$$m=\frac{y_2 - y_1}{x_2 - x_1}$$\nThis formula calculates the "rise" over the "run" or the change in $y$ divided by the change in $x$.\n\n3. **Substitute the points:** Here, $(x_1,y_1) = (5,-1)$ and $(x_2,y_2) = (-1,5)$. Substitute these values into the formula:\n$$m=\frac{5 - (-1)}{-1 - 5}$$\n\n4. **Simplify the numerator and denominator:**\n$$m=\frac{5 + 1}{-1 - 5} = \frac{6}{-6}$$\n\n5. **Cancel common factors:**\n$$m=\frac{\cancel{6}}{\cancel{-6}} = -1$$\n\n6. **Final answer:** The slope of the line between the points $(5,-1)$ and $(-1,5)$ is $-1$.