Decide whether the lines are parallel, perpendicular, or intersecting but not perpendicular.  Show your work and and explain your answer.

Line 1: 2y - x = 2

Line 2: y + 2x = 4

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?

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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!

Decide whether the lines are parallel, perpendicular, or intersecting but not perpendicular. Show your work and explain your answer.

Line 1: 2y - x = 2

Line 2: y + 2x = 4

Learn how to decide if two lines are parallel, perpendicular, or intersecting but not perpendicular using slope-intercept form. This example shows step-by-step how to find slopes and apply key concepts of linear equations. Suitable for students in Grades 8-10.

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