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To determine which lines are parallel, perpendicular, or neither relative to line \( a \) (with the equation \( y = -2x + 3 \)), let’s analyze each provided equation:

1. **Identify the slope of line \( a \):**
   - The equation of line \( a \) is \( y = -2x + 3 \), which is in slope-intercept form \( y = mx + b \).
   - Here, the slope \( m \) of line \( a \) is \( -2 \).

2. **Determine criteria for parallel and perpendicular lines:**
   - **Parallel lines** have the same slope. So, any line parallel to \( a \) must have a slope of \( -2 \).
   - **Perpendicular lines** have slopes that are negative reciprocals of each other. Thus, a line perpendicular to \( a \) would have a slope of \( \frac{1}{2} \) (the negative reciprocal of \( -2 \)).

3. **Analyze each given equation:**
   - **Equation 1:** \( y = 2x - 1 \) has a slope of \( 2 \). This is neither parallel nor perpendicular to \( a \).
   - **Equation 2:** \( y = -2x + 5 \) has a slope of \( -2 \), which is the same as line \( a \), so it is parallel to \( a \).
   - **Equation 3:** \( y = \frac{1}{2}x + 7 \) has a slope of \( \frac{1}{2} \), which is the negative reciprocal of \( -2 \), making it perpendicular to \( a \).

4. **Final classification:**
   - **Parallel to line \( a \):** \( y = -2x + 5 \)
   - **Perpendicular to line \( a \):** \( y = \frac{1}{2}x + 7 \)
   - **Neither parallel nor perpendicular to line \( a \):** \( y = 2x - 1 \)

Would you like a deeper explanation of parallel and perpendicular slope concepts?

### Related Questions
1. How do you find the slope of a line from its equation?
2. What are the properties of lines that are parallel to each other?
3. Why do perpendicular lines have slopes that are negative reciprocals?
4. Can two lines have the same slope and still not be parallel?
5. How would these concepts change in three-dimensional space?

### Tip:
When analyzing whether two lines are parallel or perpendicular, always convert the equations to slope-intercept form (\( y = mx + b \)) to easily compare slopes.

How do these equations compare to line a? Classify them as parallel, perpendicular, or neither to line a based on their slopes.

Learn how to determine if equations are parallel, perpendicular, or neither relative to a given line. This example uses the line equation y = -2x + 3 and analyzes different equations based on slope criteria, perfect for students in Grades 8-10.

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