We are given the two equations of the lines:

1. $x + 3y = 2$
2. $3x + 9y = 18$

To determine whether the lines are parallel, perpendicular, or neither, we need to find the slopes of these lines. We can express each equation in slope-intercept form, $y = mx + b$, where $m$ represents the slope.

Step 1: Put the equations in slope-intercept form.

For the first equation: $x + 3y = 2$

Rearrange to solve for $y$:

$3y = -x + 2$

$y = -\frac{1}{3}x + \frac{2}{3}$

The slope of the first line is $m_1 = -\frac{1}{3}$.

For the second equation: $3x + 9y = 18$

Rearrange to solve for $y$:

$9y = -3x + 18$

$y = -\frac{1}{3}x + 2$

The slope of the second line is $m_2 = -\frac{1}{3}$.

Step 2: Compare the slopes.

• The slopes of both lines are equal: $m_1 = m_2 = -\frac{1}{3}$.
• Since the slopes are the same, the lines are parallel.

Final Answer:

The lines are parallel.

Would you like any additional details or clarification on this? Here's some further exploration:

1. How do the slopes of parallel and perpendicular lines differ?
2. What would happen if the slopes were negative reciprocals of each other?
3. Can we determine if two lines are parallel by looking at their equations in standard form?
4. How do we find the point of intersection of two parallel lines, if any?
5. What role does the y-intercept play in determining the relationship between two lines?

Tip: To check if two lines are parallel, simply compare their slopes. If they are the same, the lines are parallel.