Determine whether the pair of lines are parallel, perpendicular, or neither. $$ \left\{\begin{array}{l}x-3 y=4 \ y=1-3 x\end{array}\right. $$

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To determine whether the pair of lines are **parallel**, **perpendicular**, or **neither**, we can analyze their slopes.

Given the equations:

1. $$ x - 3y = 4 $$
2. $$ y = 1 - 3x $$

**Step 1: Find the Slope of Each Line**

Convert both equations to the slope-intercept form $$ y = mx + b $$, where $$ m $$ is the slope.

1. **First Line: $$ x - 3y = 4 $$**
   $$
   x - 3y = 4 \
   -3y = -x + 4 \
   y = \frac{1}{3}x - \frac{4}{3}
   $$
   - **Slope ($$ m_1 $$)**: $$ \frac{1}{3} $$

2. **Second Line: $$ y = 1 - 3x $$**
   $$
   y = -3x + 1
   $$
   - **Slope ($$ m_2 $$)**: $$ -3 $$

**Step 2: Compare the Slopes**

- **Parallel Lines:** If $$ m_1 = m_2 $$, the lines are parallel.
- **Perpendicular Lines:** If $$ m_1 \times m_2 = -1 $$, the lines are perpendicular.
- **Neither:** If neither condition is met.

**Calculating the Product of Slopes:**
$$
m_1 \times m_2 = \frac{1}{3} \times (-3) = -1
$$
Since the product of the slopes is $$ -1 $$, the lines are **perpendicular**.

**Conclusion:** The given pair of lines are **perpendicular**.

**Answer:** The two lines are perpendicular to one another.
The two lines are perpendicular to each other.

Determine whether the pair of lines are parallel, perpendicular, or neither. \[ \left\{\begin{array}{l}x-3 y=4 \\ y=1-3 x\end{array}\right. \]

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