The equations of three lines are given below. Line 1: $$ 4 y=3 x+5 $$ Line 2: $$ 6 x+8 y=6 $$ Line 3: $$ y=\frac{4}{3} x-3 $$ For each pair of lines, determine whether they are parallel, perpendicular, or neither.

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1. **Rewrite each equation in slope-intercept form $$\left(y = mx + b\right)$$:**

- **Line 1:** 
     $$
     4y = 3x + 5 \quad \Rightarrow \quad y = \frac{3}{4}x + \frac{5}{4}
     $$
     The slope is $$\, m_1 = \frac{3}{4}$$.

- **Line 2:**
     $$
     6x + 8y = 6 \quad \Rightarrow \quad 8y = -6x + 6 \quad \Rightarrow \quad y = -\frac{3}{4}x + \frac{6}{8} = -\frac{3}{4}x + \frac{3}{4}
     $$
     The slope is $$\, m_2 = -\frac{3}{4}$$.

- **Line 3:**
     $$
     y = \frac{4}{3}x - 3
     $$
     The slope is $$\, m_3 = \frac{4}{3}$$.

2. **Compare each pair of lines:**

- **Line 1 and Line 2:**
     - Slopes: $$\, m_1 = \frac{3}{4}$$ and $$\, m_2 = -\frac{3}{4}$$.
     - They are **not parallel** since $$\frac{3}{4} \neq -\frac{3}{4}$$.
     - To check if they are perpendicular, verify if:
       $$
       m_1 \times m_2 = -1 \quad \Rightarrow \quad \frac{3}{4} \times \left(-\frac{3}{4}\right) = -\frac{9}{16} \neq -1.
       $$
       Thus, they are **not perpendicular**.

- **Line 1 and Line 3:**
     - Slopes: $$\, m_1 = \frac{3}{4}$$ and $$\, m_3 = \frac{4}{3}$$.
     - They are **not parallel** since $$\frac{3}{4} \neq \frac{4}{3}$$.
     - Check for perpendicularity:
       $$
       m_1 \times m_3 = \frac{3}{4} \times \frac{4}{3} = 1 \neq -1.
       $$
       So, they are **not perpendicular**.

- **Line 2 and Line 3:**
     - Slopes: $$\, m_2 = -\frac{3}{4}$$ and $$\, m_3 = \frac{4}{3}$$.
     - They are **not parallel** since $$-\frac{3}{4} \neq \frac{4}{3}$$.
     - Check for perpendicularity:
       $$
       m_2 \times m_3 = -\frac{3}{4} \times \frac{4}{3} = -1.
       $$
       Since the product is $$-1$$, **Line 2 and Line 3 are perpendicular.**

3. **Final Answers:**
   - **Line 1 and Line 2:** Neither parallel nor perpendicular.
   - **Line 1 and Line 3:** Neither parallel nor perpendicular.
   - **Line 2 and Line 3:** Perpendicular.
- **Line 1 and Line 2:** Neither parallel nor perpendicular. - **Line 1 and Line 3:** Neither parallel nor perpendicular. - **Line 2 and Line 3:** Perpendicular.

The equations of three lines are given below. Line 1: \( 4 y=3 x+5 \) Line 2: \( 6 x+8 y=6 \) Line 3: \( y=\frac{4}{3} x-3 \) For each pair of lines, determine whether they are parallel, perpendicular, or neither.

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