To find the equation of the line passing through the points $(5, -5)$ and $(-8, 8)$, we can follow these steps:

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

The formula to calculate the slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Substituting the points $(x_1, y_1) = (5, -5)$ and $(x_2, y_2) = (-8, 8)$:

$m = \frac{8 - (-5)}{-8 - 5} = \frac{8 + 5}{-8 - 5} = \frac{13}{-13} = -1$

So, the slope of the line is $m = -1$.

2. Use the point-slope form to find the equation:

The point-slope form of the equation of a line is:

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

We can use one of the points, say $(5, -5)$, and the slope $m = -1$:

$y - (-5) = -1(x - 5)$

Simplifying this:

$y + 5 = -1(x - 5)$

$y + 5 = -x + 5$

Now, subtract 5 from both sides:

$y = -x$

Conclusion:

The equation of the line is:

$\boxed{y = -x}$

Would you like further details on how to approach this problem, or do you have any other questions?

Related Questions:

1. How do I find the equation of a line given the slope and a point?
2. How do you interpret the slope of a line in real-life contexts?
3. What are the different forms of linear equations?
4. How can I verify if a point lies on a line?
5. How do you calculate the distance between two points?

Tip:

Remember, the slope represents how steep a line is, and it tells you how much $y$ changes for every unit change in $x$. If the slope is negative, the line goes down as you move from left to right!