Given the two points, (-5,3) and (-4,1), I can calculate several things based on these points. Let’s find some fundamental values:

1. Slope of the line through the points:

The slope $m$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the values: $m = \frac{1 - 3}{-4 - (-5)} = \frac{-2}{1} = -2$

2. Equation of the line:

Using the point-slope formula: $y - y_1 = m(x - x_1)$ Substitute one point (e.g., (-5, 3)) and the slope $m = -2$: $y - 3 = -2(x + 5)$ Simplify: $y = -2x - 10 + 3$ $y = -2x - 7$

3. Distance between the points:

The distance $d$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substitute the values: $d = \sqrt{(-4 - (-5))^2 + (1 - 3)^2} = \sqrt{(1)^2 + (-2)^2}$ $d = \sqrt{1 + 4} = \sqrt{5}$

4. Midpoint of the segment:

The midpoint $M$ is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Substitute the values: $M = \left( \frac{-5 + (-4)}{2}, \frac{3 + 1}{2} \right) = \left( \frac{-9}{2}, 2 \right)$ $M = \left( -4.5, 2 \right)$

Summary of Results:

1. Slope: $m = -2$
2. Equation of the line: $y = -2x - 7$
3. Distance between the points: $\sqrt{5}$
4. Midpoint: $(-4.5, 2)$

Would you like further details or have any specific questions? Here are some related questions:

1. What is the perpendicular slope for the line passing through these points?
2. How would the equation change if the slope was positive instead of negative?
3. Can we find another point that lies on the same line?
4. What is the y-intercept of the line?
5. How would you calculate the area of a triangle with these points and the origin?

Tip: Always visualize points and slopes on a graph to better understand the relationship between them.