To determine if the lines are parallel, perpendicular, or neither, we need to compare their slopes. Here's the process:

1. Equation of the first line:
   The first line is given as $y = \frac{2}{3}x + 3$.
   This equation is already in slope-intercept form $y = mx + b$, where $m$ is the slope.
   So, the slope of the first line is $m_1 = \frac{2}{3}$.

2. Equation of the second line:
   The second line is given as $3x + 2y = 6$.
   To find the slope, we first need to rearrange this into slope-intercept form $y = mx + b$.

   Start by solving for $y$: $3x + 2y = 6$ Subtract $3x$ from both sides: $2y = -3x + 6$ Now, divide everything by 2: $y = -\frac{3}{2}x + 3$ So, the slope of the second line is $m_2 = -\frac{3}{2}$.

Comparison of Slopes:

• The slopes of the two lines are:
  • $m_1 = \frac{2}{3}$
  • $m_2 = -\frac{3}{2}$

3. Parallel:
   Lines are parallel if their slopes are equal. Since $\frac{2}{3} \neq -\frac{3}{2}$, the lines are not parallel.

4. Perpendicular:
   Lines are perpendicular if the product of their slopes is $-1$. Let's check: $m_1 \times m_2 = \frac{2}{3} \times -\frac{3}{2} = -1$ Since the product is $-1$, the lines are perpendicular.

Conclusion:

The lines are perpendicular.

Would you like more details on this? Let me know if you have any questions!

Here are 5 related questions:

1. How can we determine if two lines are parallel in general?
2. What does the slope represent in a line equation?
3. How would the lines look graphically if they were parallel?
4. Can two vertical lines be perpendicular?
5. How do we find the equation of a line given a point and a slope?

Tip: To identify perpendicular lines, always check if the product of their slopes equals $-1$.