To find the equation of the line passing through the points $(5, -1)$ and $(-5, 5)$, we can follow these steps:

1. Find the slope ($m$) of the line:

The slope of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the given points $(x_1, y_1) = (5, -1)$ and $(x_2, y_2) = (-5, 5)$:

$m = \frac{5 - (-1)}{-5 - 5} = \frac{5 + 1}{-10} = \frac{6}{-10} = -\frac{3}{5}$

2. Use the point-slope form of the equation of a line:

The point-slope form of a line's equation is:

$y - y_1 = m(x - x_1)$

Using the slope $m = -\frac{3}{5}$ and the point $(5, -1)$:

$y - (-1) = -\frac{3}{5}(x - 5)$

Simplifying:

$y + 1 = -\frac{3}{5}(x - 5)$

3. Distribute and simplify:

$y + 1 = -\frac{3}{5}x + 3$

Subtract 1 from both sides to get the equation in slope-intercept form:

$y = -\frac{3}{5}x + 2$

Final Equation:

The equation of the line passing through the points $(5, -1)$ and $(-5, 5)$ is:

$\boxed{y = -\frac{3}{5}x + 2}$

Would you like further details on this process or have any other questions?

Related Questions:

1. How do you find the slope of a line given two points?
2. What is the point-slope form of a linear equation?
3. How do you convert the point-slope form to the slope-intercept form?
4. What does the slope of a line represent in a graph?
5. How can you determine if two lines are parallel?

Tip:

Always double-check your slope calculation, as it's crucial for determining the correct equation of the line.