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To find the slope of a line perpendicular to the line given by the equation \(3x + y = 8\), we can follow these steps:

1. **Rewrite the equation in slope-intercept form ( \(y = mx + b\) ):**

   Starting with:
   \[
   3x + y = 8
   \]
   Subtract \(3x\) from both sides to isolate \(y\):
   \[
   y = -3x + 8
   \]
   Here, we see that the slope (\(m\)) of the given line is \(-3\).

2. **Find the perpendicular slope:**

   The slope of a line perpendicular to another line is the negative reciprocal of the original slope. So, the negative reciprocal of \(-3\) is:
   \[
   \frac{1}{3}
   \]

Therefore, the slope of a line perpendicular to the given line is \(\frac{1}{3}\).

**Answer: \(\frac{1}{3}\)**

Would you like a detailed explanation of each step or have any questions?

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Here are five related questions to deepen understanding:
1. How do you find the slope of a parallel line instead of a perpendicular line?
2. What is the negative reciprocal, and why is it used for perpendicular lines?
3. How would the slope change if the equation were in a different form, like standard form?
4. Can you determine the y-intercept of the original line from the slope-intercept form?
5. How can you verify if two lines are perpendicular given their equations?

**Tip:** Remember, for perpendicular lines, the product of their slopes should equal \(-1\).

Find the slope of a line perpendicular to the line whose equation is 3x + y = 8. Fully simplify your answer.

Learn how to find the slope of a line perpendicular to the line given by the equation 3x + y = 8. This problem covers algebraic concepts, including linear equations and the slopes of perpendicular lines. Suitable for students in Grades 8-10.

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