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. **Writing equations in slope-intercept form**

- **Line 1:** Given $$4y = 3x + 5$$, divide both sides by 4:
     
     $$
     y = \frac{3}{4}x + \frac{5}{4}
     $$
     
     The slope of Line 1 is $$m_1 = \frac{3}{4}$$.

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

- **Line 3:** Given $$y = \frac{4}{3}x - 3$$, it is already in slope-intercept form with slope $$m_3 = \frac{4}{3}$$.

2. **Comparing pairs of lines**

- **Line 1 and Line 2:**
     
     - Slopes: $$m_1 = \frac{3}{4}$$ and $$m_2 = -\frac{3}{4}$$.
     - They are not equal, so the lines are not parallel.
     - Check for perpendicularity: compute the product:
       
       $$
       m_1 \cdot m_2 = \frac{3}{4} \times \left(-\frac{3}{4}\right) = -\frac{9}{16}
       $$
       
       Since $$-\frac{9}{16} \neq -1$$, the lines are not perpendicular.

- **Line 1 and Line 3:**
     
     - Slopes: $$m_1 = \frac{3}{4}$$ and $$m_3 = \frac{4}{3}$$.
     - They are not equal, so the lines are not parallel.
     - Check for perpendicularity:
       
       $$
       m_1 \cdot m_3 = \frac{3}{4} \times \frac{4}{3} = 1
       $$
       
       Since $$1 \neq -1$$, the lines are not perpendicular.

- **Line 2 and Line 3:**
     
     - Slopes: $$m_2 = -\frac{3}{4}$$ and $$m_3 = \frac{4}{3}$$.
     - They are not equal, so the lines are not parallel.
     - Check for perpendicularity:
       
       $$
       m_2 \cdot m_3 = -\frac{3}{4} \times \frac{4}{3} = -1
       $$
       
       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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