We are given two lines, and we need to determine whether they are parallel, perpendicular, or intersecting but not perpendicular.

Step 1: Write each equation in slope-intercept form (i.e., $y = mx + b$, where $m$ is the slope).

For both lines, let's solve for $y$.

Line 1: $2y - x = 2$

Solve for $y$: $2y = x + 2$ $y = \frac{1}{2}x + 1$ So, the slope of Line 1 is $m_1 = \frac{1}{2}$.

Line 2: $y + 2x = 4$

Solve for $y$: $y = -2x + 4$ So, the slope of Line 2 is $m_2 = -2$.

Step 2: Analyze the slopes.

• If two lines are parallel, their slopes will be equal: $m_1 = m_2$.
• If two lines are perpendicular, their slopes will be negative reciprocals of each other: $m_1 \times m_2 = -1$.
• If two lines are neither parallel nor perpendicular, they will intersect but not at a right angle.

Step 3: Compare the slopes.

• The slope of Line 1 is $m_1 = \frac{1}{2}$.
• The slope of Line 2 is $m_2 = -2$.

Now, check if the lines are perpendicular: $m_1 \times m_2 = \frac{1}{2} \times (-2) = -1$ Since $m_1 \times m_2 = -1$, the lines are perpendicular.

Final Answer:

The two lines are perpendicular.

Would you like more details on how slopes determine the relationship between lines, or have any other questions?

---

Here are 5 related questions:

1. How do you determine if two lines are parallel from their equations?
2. What is the geometric meaning of perpendicular lines?
3. Can two lines with the same y-intercept ever be perpendicular?
4. How do we find the point of intersection between two non-parallel lines?
5. How does changing the slope affect the orientation of a line?

Tip: When finding slopes, always make sure the equation is in slope-intercept form $y = mx + b$ to easily identify the slope!