To determine the slopes requested, let’s analyze the given line equation and work through the calculations:

The equation of the line is: $x + 6y = 4$

1. Rewrite the equation in slope-intercept form (i.e., $y = mx + b$) to identify the slope ($m$): $x + 6y = 4$ Subtract $x$ from both sides: $6y = -x + 4$ Now, divide by 6: $y = -\frac{1}{6}x + \frac{2}{3}$

   From this, we see the slope $m$ of the given line is $-\frac{1}{6}$.

2. Slope of a line parallel to this line: A line parallel to the given line will have the same slope. $\text{Slope of parallel line} = -\frac{1}{6}$

3. Slope of a line perpendicular to this line: The slope of a line perpendicular to this one is the negative reciprocal of $-\frac{1}{6}$. $\text{Slope of perpendicular line} = 6$

Answers:

• Slope of a parallel line: $-\frac{1}{6}$
• Slope of a perpendicular line: $6$

Would you like further details on this, or do you have any questions?

Here are five related questions to expand on this topic:

1. How do you find the slope of a line from its equation in standard form?
2. Why is the slope of a perpendicular line the negative reciprocal?
3. What are some examples of applications of parallel and perpendicular lines in real life?
4. How can we determine if two lines are parallel or perpendicular just from their equations?
5. How does changing the constant in the equation affect the line's position but not its slope?

Tip: When dealing with line equations, always aim to get the equation into slope-intercept form to easily identify the slope.