9. Are the following lines parallel, perpendicular or neither? a) $$ y=3 x+2 $$ and $$ y=-\frac{1}{3} x+4 $$ b) $$ y=\frac{1}{2} x-1 $$ and $$ 2 y-x=2 $$ c) $$ 3 y-x=9 $$ and $$ y=3 x-7 $$ d) $$ 2 y+3 x=8 $$ and $$ 3 y=2 x+1 $$

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To determine whether each pair of lines is **parallel**, **perpendicular**, or **neither**, we'll analyze their slopes.

- **Parallel Lines** have identical slopes.
- **Perpendicular Lines** have slopes that are negative reciprocals of each other (i.e., $$ m_1 	imes m_2 = -1 $$).
- If neither condition is met, the lines are **neither** parallel nor perpendicular.

Let's examine each pair:

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### **a) $$ y = 3x + 2 $$ and $$ y = -\frac{1}{3}x + 4 $$**

1. **Identify the slopes:**
   - **First line:** $$ y = 3x + 2 $$ has a slope $$ m_1 = 3 $$.
   - **Second line:** $$ y = -\frac{1}{3}x + 4 $$ has a slope $$ m_2 = -\frac{1}{3} $$.

2. **Check for Parallelism:**
   - $$ m_1 
eq m_2 $$ $$ \Rightarrow $$ Not parallel.

3. **Check for Perpendicularity:**
   - $$ m_1 	imes m_2 = 3 	imes \left(-\frac{1}{3}ight) = -1 $$.
   - $$ m_1 	imes m_2 = -1 $$ $$ \Rightarrow $$ Perpendicular.

**Answer:** **Perpendicular**

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### **b) $$ y = \frac{1}{2}x - 1 $$ and $$ 2y - x = 2 $$**

1. **Rearrange the second equation to slope-intercept form $$ y = mx + b $$:**
   - $$ 2y = x + 2 $$
   - $$ y = \frac{1}{2}x + 1 $$

2. **Identify the slopes:**
   - **First line:** $$ y = \frac{1}{2}x - 1 $$ has a slope $$ m_1 = \frac{1}{2} $$.
   - **Second line:** $$ y = \frac{1}{2}x + 1 $$ has a slope $$ m_2 = \frac{1}{2} $$.

3. **Check for Parallelism:**
   - $$ m_1 = m_2 $$ $$ \Rightarrow $$ Parallel.

4. **Check for Perpendicularity:**
   - Not applicable since slopes are equal.

**Answer:** **Parallel**

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### **c) $$ 3y - x = 9 $$ and $$ y = 3x - 7 $$**

1. **Rearrange the first equation to slope-intercept form:**
   - $$ 3y = x + 9 $$
   - $$ y = \frac{1}{3}x + 3 $$

2. **Identify the slopes:**
   - **First line:** $$ y = \frac{1}{3}x + 3 $$ has a slope $$ m_1 = \frac{1}{3} $$.
   - **Second line:** $$ y = 3x - 7 $$ has a slope $$ m_2 = 3 $$.

3. **Check for Parallelism:**
   - $$ m_1 
eq m_2 $$ $$ \Rightarrow $$ Not parallel.

4. **Check for Perpendicularity:**
   - $$ m_1 	imes m_2 = \frac{1}{3} 	imes 3 = 1 
eq -1 $$ $$ \Rightarrow $$ Not perpendicular.

**Answer:** **Neither**

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### **d) $$ 2y + 3x = 8 $$ and $$ 3y = 2x + 1 $$**

1. **Rearrange both equations to slope-intercept form:**
   - **First equation:** $$ 2y + 3x = 8 $$ 
     - $$ 2y = -3x + 8 $$
     - $$ y = -\frac{3}{2}x + 4 $$
   - **Second equation:** $$ 3y = 2x + 1 $$
     - $$ y = \frac{2}{3}x + \frac{1}{3} $$

2. **Identify the slopes:**
   - **First line:** $$ y = -\frac{3}{2}x + 4 $$ has a slope $$ m_1 = -\frac{3}{2} $$.
   - **Second line:** $$ y = \frac{2}{3}x + \frac{1}{3} $$ has a slope $$ m_2 = \frac{2}{3} $$.

3. **Check for Parallelism:**
   - $$ m_1 
eq m_2 $$ $$ \Rightarrow $$ Not parallel.

4. **Check for Perpendicularity:**
   - $$ m_1 	imes m_2 = -\frac{3}{2} 	imes \frac{2}{3} = -1 $$.
   - $$ m_1 	imes m_2 = -1 $$ $$ \Rightarrow $$ Perpendicular.

**Answer:** **Perpendicular**

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### **Summary of Results:**

- **a)** Perpendicular
- **b)** Parallel
- **c)** Neither
- **d)** Perpendicular
a) Perpendicular b) Parallel c) Neither d) Perpendicular

9. Are the following lines parallel, perpendicular or neither? a) \( y=3 x+2 \) and \( y=-\frac{1}{3} x+4 \) b) \( y=\frac{1}{2} x-1 \) and \( 2 y-x=2 \) c) \( 3 y-x=9 \) and \( y=3 x-7 \) d) \( 2 y+3 x=8 \) and \( 3 y=2 x+1 \)

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