Question

Which equations represent the line that is perpendicular to the line 5x − 2y = −6 and passes through the point (5, −4)? Select three options.

y = –Two-fifthsx – 2
2x + 5y = −10
2x − 5y = −10
y + 4 = –Two-fifths(x – 5)
y – 4 = Five-halves(x + 5)

Answer 1

To find the equations that represent a line perpendicular to the line given by the equation \(5x - 2y = -6\) and passes through the point \((5, -4)\), we first need to determine the slope of the original line.

1. **Convert \(5x - 2y = -6\) to slope-intercept form (y = mx + b)**:

\[
-2y = -5x - 6
\]
\[
y = \frac{5}{2}x + 3
\]

The slope \(m\) of the original line is \(\frac{5}{2}\).

2. **Find the slope of the perpendicular line**:
The slope of a line that is perpendicular to another is the negative reciprocal of the original line's slope. Thus, the slope of the perpendicular line is:

\[
m_{\text{perpendicular}} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5}
\]

3. **Use the point-slope form of a line to find the equation of the perpendicular line that passes through \((5, -4)\)**:
The point-slope form is given by:

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

Plugging in the point \((5, -4)\) and the slope \(-\frac{2}{5}\):

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

4. **Identify the options**:

**Let's analyze the provided options:**

- **Option 1: \(y = -\frac{2}{5}x - 2\)**: This does not match our point-slope form.
- **Option 2: \(2x + 5y = -10\)**: Rearranging this gives \(5y = -2x - 10\) or \(y = -\frac{2}{5}x - 2\) (which is wrong, so this does NOT represent the line we want).
- **Option 3: \(2x - 5y = -10\)**: Rearranging gives \(5y = 2x + 10\) or \(y = \frac{2}{5}x + 2\) (which is incorrect, so this does NOT represent the line we want).
- **Option 4: \(y + 4 = -\frac{2}{5}(x - 5)\)**: This matches directly our derived equation (which is correct).
- **Option 5: \(y - 4 = \frac{5}{2}(x + 5)\)**: This has a slope of \(\frac{5}{2}\) and does not match our needed slope.

**Final Selection**:
The only correct option representing the required line is:
- \(y + 4 = -\frac{2}{5}(x - 5)\).

None of the other equations are correct as they either do not have the correct slope or do not pass through the given point.

So, the only correct choice is:
- \(y + 4 = -\frac{2}{5}(x - 5)\).