Question

Select the equation that is perpendicular to 5x-4y=16

a
y=(4/3)x+4

b
y=(5/4)x+6

c
y=(-4/5)x-2

Answer 1

To find the equation that is perpendicular to the given equation \(5x - 4y = 16\), we first need to determine the slope of the given line.

We can rearrange the equation into slope-intercept form \(y = mx + b\):

1. Start with the original equation:
\[
5x - 4y = 16
\]

2. Solve for \(y\):
\[
-4y = -5x + 16
\]
\[
y = \frac{5}{4}x - 4
\]

From this, we can see that the slope \(m\) of the line is \(\frac{5}{4}\).

The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope \(m_{\perp}\) of the perpendicular line is:
\[
m_{\perp} = -\frac{1}{\left(\frac{5}{4}\right)} = -\frac{4}{5}
\]

Now, let's check the slopes of each of the given options:

a) \(y = \frac{4}{3}x + 4\) has a slope of \(\frac{4}{3}\) (not perpendicular).

b) \(y = \frac{5}{4}x + 6\) has a slope of \(\frac{5}{4}\) (not perpendicular).

c) \(y = -\frac{4}{5}x - 2\) has a slope of \(-\frac{4}{5}\) (perpendicular).

Therefore, the correct answer is:

**c) \(y = -\frac{4}{5}x - 2\)**.