Question

A) Which line is perpendicular to the line and goes through the point ?
(1 point)
5x − 4y = 12 (5, 2)
y = − x − 3
4
5
y = − x + 6
4
5
y = x + 6
4
5
y = x + 6

Answer 1

To find a line that is perpendicular to the given line \(5x - 4y = 12\) and goes through the point \((5, 2)\), we can follow these steps:

1. **Find the slope of the given line:**
First, rewrite the equation \(5x - 4y = 12\) in slope-intercept form (y = mx + b).
\[
-4y = -5x + 12
\]
\[
y = \frac{5}{4}x - 3
\]
The slope of this line (m) is \(\frac{5}{4}\).

2. **Determine the slope of the perpendicular line:**
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Thus, we find the perpendicular slope:
\[
m_{\text{perpendicular}} = -\frac{1}{\frac{5}{4}} = -\frac{4}{5}
\]

3. **Use the point-slope form of the line equation:**
Now that we have the slope of the perpendicular line and the point \((5, 2)\), we can use the point-slope form to find the equation of the perpendicular line:
\[
y - y_1 = m(x - x_1)
\]
Substituting \(m = -\frac{4}{5}\), \(x_1 = 5\), and \(y_1 = 2\):
\[
y - 2 = -\frac{4}{5}(x - 5)
\]
Simplifying:
\[
y - 2 = -\frac{4}{5}x + 4
\]
\[
y = -\frac{4}{5}x + 6
\]

4. **Finding the matching line equation:**
The equation we have derived is:
\[
y = -\frac{4}{5}x + 6
\]

5. **Compare with the given options:**
From the options given:
- \(y = -x - 3\)
- \(y = -\frac{4}{5}x + 6\)
- \(y = x + 6\)

The line that is perpendicular to \(5x - 4y = 12\) and passes through the point \((5, 2)\) is:
\[
\boxed{y = -\frac{4}{5}x + 6}
\]