Question

On a coordinate plane, a line goes through (negative 1, 1) and (0, negative 3). A point is at (negative 4, negative 3) and (0, negative 3).
What is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point (−4, −3)?

y + 3 = −4(x + 4)
y + 3 = –One-fourth(x + 4)
y + 3 = One-fourth(x + 4)
y + 3 = 4(x + 4)

Answer 1

To find the equation of the line that is perpendicular to the given line and passes through the point \((-4, -3)\), we first need to determine the slope of the given line.

1. **Finding the slope of the given line:**
The points on the line are \((-1, 1)\) and \( (0, -3)\). We can calculate the slope \(m\) using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 1}{0 - (-1)} = \frac{-4}{1} = -4
\]

2. **Finding the slope of the perpendicular line:**
The slope of the line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, if the slope of the original line is \(-4\), the slope \(m'\) of the perpendicular line is:

\[
m' = -\frac{1}{-4} = \frac{1}{4}
\]

3. **Writing the equation in point-slope form:**
We want the equation of the line with slope \(\frac{1}{4}\) that passes through the point \((-4, -3)\). The point-slope form of the equation of a line is given by:

\[
y - y_1 = m(x - x_1)
\]

Using the point \((-4, -3)\) and the slope \(\frac{1}{4}\):

\[
y - (-3) = \frac{1}{4}(x - (-4))
\]

Simplifying this gives:

\[
y + 3 = \frac{1}{4}(x + 4)
\]

Thus, the correct answer is:
\[
\boxed{y + 3 = \frac{1}{4}(x + 4)}
\]