Consider the equation − x + 4 y

To find the slope of a line parallel or perpendicular to the given line $-x + 4y = 12$, we first need to express the equation in slope-intercept form, $y = mx + b$, where $m$ is the slope.

Step 1: Rewrite the equation in slope-intercept form.

Start with the given equation:

$-x + 4y = 12$

Add $x$ to both sides:

$4y = x + 12$

Now, divide both sides by 4 to solve for $y$:

$y = \frac{1}{4}x + 3$

Step 2: Identify the slope.

From the slope-intercept form $y = \frac{1}{4}x + 3$, the slope $m$ of the original line is $\frac{1}{4}$.

Step 3: Find the slopes of parallel and perpendicular lines.

• A line parallel to the given line will have the same slope, so the slope of a parallel line is also $\frac{1}{4}$.

• A line perpendicular to the given line will have a slope that is the negative reciprocal of $\frac{1}{4}$. The negative reciprocal of $\frac{1}{4}$ is $-4$.

Final Answer:

• The slope of a line parallel to the given line is $\frac{1}{4}$.
• The slope of a line perpendicular to the given line is $-4$.

Would you like further details or clarifications?

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Here are 5 related questions for further practice:

1. What is the y-intercept of the line $-x + 4y = 12$?
2. How do you find the equation of a line parallel to $-x + 4y = 12$ passing through the point (2, 5)?
3. What is the equation of a line perpendicular to $-x + 4y = 12$ and passing through (1, -3)?
4. How do you determine the distance between two parallel lines?
5. What is the significance of the negative reciprocal in finding perpendicular slopes?

Tip: Always convert the line to slope-intercept form $y = mx + b$ to easily identify the slope.