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The problem presented asks for the slopes of two lines, then asks whether they are parallel, perpendicular, or neither. The equations are:

1. \( 15x - 2y = 6 \)
2. \( 2x + 15y = 60 \)

### Step 1: Convert both equations to slope-intercept form (\( y = mx + b \)), where \( m \) represents the slope.

#### Equation 1: \( 15x - 2y = 6 \)

Solve for \( y \):
\[
15x - 2y = 6
\]
\[
-2y = -15x + 6
\]
\[
y = \frac{15}{2}x - 3
\]
The slope of the first line (\( m_1 \)) is \( \frac{15}{2} \).

#### Equation 2: \( 2x + 15y = 60 \)

Solve for \( y \):
\[
2x + 15y = 60
\]
\[
15y = -2x + 60
\]
\[
y = -\frac{2}{15}x + 4
\]
The slope of the second line (\( m_2 \)) is \( -\frac{2}{15} \).

### Step 2: Determine the relationship between the lines
- **Parallel** lines have equal slopes.
- **Perpendicular** lines have slopes that are negative reciprocals, meaning \( m_1 \cdot m_2 = -1 \).

Let's check:

\[
m_1 = \frac{15}{2}, \quad m_2 = -\frac{2}{15}
\]

Now, multiply the slopes:

\[
m_1 \cdot m_2 = \frac{15}{2} \cdot \left(-\frac{2}{15}\right) = -1
\]

Since \( m_1 \cdot m_2 = -1 \), the lines are **perpendicular**.

Would you like more details on any step or have questions? Here are some related questions for further practice:

1. How do you find the slope of a line given its equation?
2. What is the condition for two lines to be parallel?
3. How can you tell if two lines are neither parallel nor perpendicular?
4. What is the slope of a vertical line?
5. How do you interpret the y-intercept in slope-intercept form?

**Tip:** Always convert equations to slope-intercept form to easily compare the slopes.

For the given pair of equations, give the slopes of the lines, and then determine whether the two lines are parallel, perpendicular, or neither. The equations are: 15x - 2y = 6 and 2x + 15y = 60.

This problem requires converting two linear equations (15x - 2y = 6 and 2x + 15y = 60) into slope-intercept form to find the slopes and determine whether the lines are parallel, perpendicular, or neither. The solution involves using the slope-intercept form and checking for the negative reciprocal relationship between slopes.

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