Question

Write an equation of the line that passes through (−3,−1)
and is perpendicular to the line y=−19x−6
.



Write a perpendicular line: Find Equation of Line perpendicular to a line through given point

Answer 1

To find the equation of the line that passes through the point \((-3, -1)\) and is perpendicular to the line given by the equation \(y = -19x - 6\), we first need to determine the slope of the given line.

The slope-intercept form of the line \(y = mx + b\) gives us the slope \(m\). In this case, the slope of the line \(y = -19x - 6\) is:

\[
m_1 = -19
\]

The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. Therefore, the slope \(m_2\) of the perpendicular line is:

\[
m_2 = -\frac{1}{m_1} = -\frac{1}{-19} = \frac{1}{19}
\]

Now that we have the slope of the perpendicular line, we can use the point-slope form of the equation of a line, which is:

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

where \((x_1, y_1)\) is the point through which the line passes. Substituting \((-3, -1)\) and \(m = \frac{1}{19}\) into the equation gives us:

\[
y - (-1) = \frac{1}{19}(x - (-3))
\]

This simplifies to:

\[
y + 1 = \frac{1}{19}(x + 3)
\]

Next, we can distribute \(\frac{1}{19}\):

\[
y + 1 = \frac{1}{19}x + \frac{3}{19}
\]

Now, we will isolate \(y\):

\[
y = \frac{1}{19}x + \frac{3}{19} - 1
\]

To combine the constants, we convert \(1\) to a fraction:

\[
1 = \frac{19}{19}
\]

Thus,

\[
y = \frac{1}{19}x + \frac{3}{19} - \frac{19}{19} = \frac{1}{19}x + \frac{3 - 19}{19} = \frac{1}{19}x - \frac{16}{19}
\]

So, the equation of the line that is perpendicular to \(y = -19x - 6\) and passes through the point \((-3, -1)\) is:

\[
y = \frac{1}{19}x - \frac{16}{19}
\]