The task requires you to write parametric equations for a line passing through two points, $(8, -6)$ and $(-8, -2)$, corresponding to the parameter $t$.

Step 1: Find the slope of the line.

To find the slope, use the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute the points $(x_1, y_1) = (8, -6)$ and $(x_2, y_2) = (-8, -2)$: $m = \frac{-2 - (-6)}{-8 - 8} = \frac{-2 + 6}{-16} = \frac{4}{-16} = -\frac{1}{4}$ The slope of the line is $m = -\frac{1}{4}$.

Step 2: Write the parametric equations.

In parametric form, the equations for the line are given by: $x = x_0 + t \cdot dx$ $y = y_0 + t \cdot dy$ where $(x_0, y_0)$ is the starting point, and $(dx, dy)$ are the changes in $x$ and $y$ with respect to $t$.

• For the $x$-coordinate, we can express $x = 8 - 4t$ because the change in $x$ (from $8$ to $-8$) is $-16$, which corresponds to a $-4$ per unit change in $t$.

• For the $y$-coordinate, using the slope of $-\frac{1}{4}$, the change in $y$ is $-1$ for every 4 units of change in $x$. This leads to the equation $y = -6 + t$.

Thus, the parametric equations for the line are: $x = 8 - 4t$ $y = -6 + t$

Would you like more details or clarification on any of the steps? Here are some related questions:

1. How would you convert these parametric equations into standard form?
2. How would you find the parametric equations for a line passing through two different points?
3. What is the relationship between the slope of a line and its parametric equations?
4. How do you calculate the direction of the line based on its parametric equations?
5. What would happen if the line was vertical? How would the parametric equations change?

Tip: Remember that the slope of a line in parametric equations can also be found by looking at the rate of change in both $x$ and $y$ with respect to the parameter $t$.