Question

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

y=−45x+6
y equals negative Start Fraction 4 over 5 End Fraction x plus 6

y=−45x−3
y equals negative Start Fraction 4 over 5 End Fraction x minus 3

y=54x+6
y equals Start Fraction 5 over 4 End Fraction x plus 6

y=45x+6

Answer 1

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

1. **Find the slope of the given line**:

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

\[
5x - 4y = 12
\]
\[
-4y = -5x + 12
\]
\[
y = \frac{5}{4}x - 3
\]

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

2. **Find the slope of the perpendicular line**:

The slope of a line that is perpendicular to another line is the negative reciprocal of the original line’s slope. Therefore, the slope of the perpendicular line is:

\[
m = -\frac{1}{\frac{5}{4}} = -\frac{4}{5}
\]

3. **Use point-slope form to find the equation of the line**:

We use the point-slope form of the equation of a line, which is:

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

where \( (x_1, y_1) = (5, 2) \) and \( m = -\frac{4}{5} \):

\[
y - 2 = -\frac{4}{5}(x - 5)
\]

Expanding this, we get:

\[
y - 2 = -\frac{4}{5}x + 4
\]
\[
y = -\frac{4}{5}x + 6
\]

Now, we can compare this result to the provided options:

- **y equals negative Start Fraction 4 over 5 End Fraction x plus 6**

Thus, the equation of the line that is perpendicular to \( 5x - 4y = 12 \) and passes through the point \( (5, 2) \) is:

**y equals negative Start Fraction 4 over 5 End Fraction x plus 6**