Which of the following equations represents the line that passes through the point (0, -6) and that is parallel to the line x - 4y - 5 = 0?


A
x
-4x - y - 6 = 0
B
x
4x - y - 6 = 0
C
x
x - 4y - 24 = 0
D
x
x - 4y + 6 = 0

since it is parallel, it will differ only in the constant, so

new equation is
x - 4y + c = 0

plug in the given point to find c.

To determine which of the given equations represents the line that passes through the point (0, -6) and is parallel to the line x - 4y - 5 = 0, we need to find the equation of the parallel line.

First, let's find the slope of the given line. The equation of the given line is x - 4y - 5 = 0. To get the equation into slope-intercept form (y = mx + b), we need to solve for y:

x - 4y - 5 = 0
-4y = -x + 5
y = (1/4)x - 5/4

The slope of the given line is 1/4.

Since the parallel line will have the same slope, we can write the equation of the parallel line in slope-intercept form as: y = (1/4)x + b

To find the value of b, we can plug the coordinates (0, -6) into the equation:

-6 = (1/4)(0) + b
-6 = b

So, the equation of the parallel line is y = (1/4)x - 6.

Now, let's check which of the given equations matches this equation:

A. -4x - y - 6 = 0
B. 4x - y - 6 = 0
C. x - 4y - 24 = 0
D. x - 4y + 6 = 0

Plugging in y = (1/4)x - 6 into each of the equations, we see that only option D results in a true equation:

0 - 4(-6) + 6 = 0
24 + 6 = 0
30 ≠ 0

Therefore, option D, x - 4y + 6 = 0, represents the line that passes through the point (0, -6) and is parallel to the line x - 4y - 5 = 0.

To find the equation of a line that is parallel to the given line and passes through the point (0, -6), we need to follow these steps:

Step 1: Determine the slope of the given line.
The given line is x - 4y - 5 = 0. To find the slope, we need to rearrange the equation to the form "y = mx + b" (where m represents the slope):
x - 4y - 5 = 0
(x - 5) - 4y = 0
-4y = -x + 5
y = (1/4)x - (5/4)

From this equation, we can see that the slope of the given line is 1/4.

Step 2: Determine the equation of the line parallel to the given line and passing through the point (0, -6).
Since the parallel line has the same slope as the given line, we can use the point-slope form of the equation:
y - y1 = m(x - x1)

Substituting the values m = 1/4, x1 = 0, and y1 = -6 into the equation, we have:
y - (-6) = (1/4)(x - 0)
y + 6 = (1/4)x

Now, rearrange the equation to the standard form:
(1/4)x - y - 6 = 0
Multiply both sides by 4 to eliminate the fraction:
x - 4y - 24 = 0

Therefore, the correct equation representing the line that passes through the point (0, -6) and is parallel to the line x - 4y - 5 = 0 is:
C) x - 4y - 24 = 0