Question

Find the line that is perpendicular to y=1/2x+4
and passes through the point (−3,4)
.

Answer 1

To find a line that is perpendicular to the given line \( y = \frac{1}{2}x + 4 \) and passes through the point \((-3, 4)\), we first need to determine the slope of the given line.

The slope of the line \( y = \frac{1}{2}x + 4 \) is \( \frac{1}{2} \).

The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. The negative reciprocal of \( \frac{1}{2} \) is \( -2 \).

Now, we have the slope of the perpendicular line, which is \( -2 \), and we need to use the point-slope form of the equation of a line, which is given by:

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

where \((x_1, y_1)\) is the point the line goes through and \(m\) is the slope. Here, \((x_1, y_1) = (-3, 4)\) and \(m = -2\).

Substituting these values into the equation:

\[
y - 4 = -2(x - (-3))
\]

Simplifying this:

\[
y - 4 = -2(x + 3)
\]
\[
y - 4 = -2x - 6
\]
\[
y = -2x - 6 + 4
\]
\[
y = -2x - 2
\]

Thus, the equation of the line that is perpendicular to \( y = \frac{1}{2}x + 4 \) and passes through the point \((-3, 4)\) is:

\[
\boxed{y = -2x - 2}
\]