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We are given parametric equations for a line as follows:

\[
x = t - 8
\]
\[
y = \frac{t}{4} - 3
\]

To find the rectangular equation (i.e., the equation in terms of \(x\) and \(y\)):

1. First, solve for \(t\) from the equation for \(x\):
   \[
   t = x + 8
   \]

2. Now substitute this expression for \(t\) into the equation for \(y\):
   \[
   y = \frac{x + 8}{4} - 3
   \]

3. Simplify the equation:
   \[
   y = \frac{x + 8}{4} - 3 = \frac{x}{4} + 2 - 3 = \frac{x}{4} - 1
   \]

Thus, the rectangular equation of the line is:
\[
y = \frac{x}{4} - 1
\]

### Graphing the Line and Showing Orientation
- The equation \(y = \frac{x}{4} - 1\) represents a line with a slope of \(\frac{1}{4}\) and a y-intercept of \(-1\).
- To indicate the direction (orientation), as \(t\) increases, \(x = t - 8\) and \(y = \frac{t}{4} - 3\) both increase. This means the line is traced in the direction of increasing \(x\) and \(y\) as \(t\) increases.

If you'd like, I can generate a graph of this line along with its orientation. Would you like that?

Here are five related questions for further practice:

1. How would the rectangular equation change if \(x = 2t - 5\) and \(y = \frac{t}{3} + 1\)?
2. What happens to the slope of the line if the parametric equations are \(x = t - 4\) and \(y = 2t - 3\)?
3. Can you convert the parametric equations \(x = t^2\) and \(y = 2t + 1\) into a rectangular equation?
4. What would be the parametric equations for a line with a slope of 2 and passing through the point (3, 1)?
5. How can you verify that a parametric equation represents a linear equation?

### Tip:
When solving parametric equations, always solve for \(t\) in terms of \(x\) or \(y\) first to make substitution easier.

Find the rectangular equation of the line given by the parametric equations: x = t - 8, y = t/4 - 3. Then graph the line and show its orientation. Assume that t can be any real number.

Learn how to convert parametric equations into rectangular form with the example x = t - 8 and y = t/4 - 3. Detailed step-by-step instructions are provided, along with the graph and orientation of the resulting line. Ideal for high school algebra students.

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