Question

Is each line parallel, perpendicular, or neither parallel nor perpendicular to the line −x+4y=20?

Drag each choice into the boxes to correctly complete the table.

A. -x + 4y = 8
B. 4x + y = -1
C. y = -1/4x + 6
D. y = -4x - 3

Answer 1

To determine whether each line is parallel, perpendicular, or neither to the line \(-x + 4y = 20\), we first need to find the slope of this line.

1. Convert \(-x + 4y = 20\) to slope-intercept form \(y = mx + b\):
\[
4y = x + 20 \\
y = \frac{1}{4}x + 5
\]
The slope \(m\) of this line is \(\frac{1}{4}\).

Now, we can analyze each option:

### A. \(-x + 4y = 8\)
Convert \(-x + 4y = 8\) to slope-intercept form:
\[
4y = x + 8 \\
y = \frac{1}{4}x + 2
\]
- Slope = \(\frac{1}{4}\) (parallel to \(-x + 4y = 20\))

### B. \(4x + y = -1\)
Convert \(4x + y = -1\) to slope-intercept form:
\[
y = -4x - 1
\]
- Slope = \(-4\) (perpendicular to \(-x + 4y = 20\), since \(-4\) is the negative reciprocal of \(\frac{1}{4}\))

### C. \(y = -\frac{1}{4}x + 6\)
Given the equation is already in slope-intercept form:
- Slope = \(-\frac{1}{4}\) (neither parallel nor perpendicular, since it is not equal to \(\frac{1}{4}\) nor is it the negative reciprocal)

### D. \(y = -4x - 3\)
Given the equation is already in slope-intercept form:
- Slope = \(-4\) (perpendicular to \(-x + 4y = 20\), since \(-4\) is the negative reciprocal of \(\frac{1}{4}\))

### Summary:
Now we can fill out the table as follows:

- A: **Parallel**
- B: **Perpendicular**
- C: **Neither**
- D: **Perpendicular**

Therefore, the table is completed:

| Line | Relationship |
|-----------------------|--------------------|
| A. -x + 4y = 8 | Parallel |
| B. 4x + y = -1 | Perpendicular |
| C. y = -1/4x + 6 | Neither |
| D. y = -4x - 3 | Perpendicular |