Let's examine the two equations provided and determine if the lines are parallel, perpendicular, or neither.

Step 1: Convert both equations to slope-intercept form $y = mx + b$, where $m$ is the slope.

1. First equation: $6y = x - 12$ Divide both sides by 6 to solve for $y$: $y = \frac{1}{6}x - 2$ So, the slope $m_1$ of the first line is $\frac{1}{6}$.

2. Second equation: $18y + 36 = 3x$ Subtract 36 from both sides: $18y = 3x - 36$ Divide both sides by 18 to solve for $y$: $y = \frac{1}{6}x - 2$ So, the slope $m_2$ of the second line is also $\frac{1}{6}$.

Step 2: Compare the slopes

• If the slopes are equal and the intercepts differ, the lines are parallel.
• If the slopes are negative reciprocals of each other (i.e., $m_1 \times m_2 = -1$), the lines are perpendicular.
• If neither of the above conditions is met, the lines are neither parallel nor perpendicular.

Here, both lines have the same slope $\frac{1}{6}$ and the same y-intercept (-2). Therefore, the lines are identical (coincident), meaning they are parallel and overlap completely.

Final Answer: The lines are parallel.

Do you need more details on this, or have any questions?

Here are five related questions to expand on this concept:

1. How do we determine if two lines are perpendicular in three dimensions?
2. What happens if two lines have the same slope but different intercepts?
3. How would you write the equation of a line perpendicular to one of these lines?
4. Can lines with undefined slopes (vertical lines) be parallel?
5. What is the geometric interpretation of coincident lines?

Tip: To easily find if two lines are perpendicular, just check if the product of their slopes equals -1!