Are the lines y= 2/3x+3 and 3x+2y=6 parallel, perpendicular or neither?

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 \(-1\).

Learn how to determine if the lines y = 2/3x + 3 and 3x + 2y = 6 are parallel, perpendicular, or neither. This problem uses algebraic concepts such as slope-intercept form and the rules for parallel and perpendicular lines. Suitable for students in Grades 8-10.

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