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?

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 $-1$.