Determine if the lines are parallel, perpendicular, or neither. Justify you
a. y = -4x + 2 and y = 4r+8

b. 5x + 15y =-8
y=-1/3xー9

To determine whether two lines are parallel, perpendicular, or neither, we compare their slopes. Here's how to approach both problems:

### (a) \( y = -4x + 2 \) and \( y = 4x + 8 \)

These are already in slope-intercept form, \( y = mx + b \), where \( m \) is the slope.

- For the first equation, \( y = -4x + 2 \), the slope is \( m_1 = -4 \).
- For the second equation, \( y = 4x + 8 \), the slope is \( m_2 = 4 \).

- **Parallel lines** have the **same slope**: \( m_1 = m_2 \).
- **Perpendicular lines** have slopes that are **negative reciprocals**: \( m_1 \times m_2 = -1 \).

Here:
- \( m_1 = -4 \) and \( m_2 = 4 \), so they are **not equal** and **not negative reciprocals**.

**Conclusion**: The lines are **neither** parallel nor perpendicular.

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### (b) \( 5x + 15y = -8 \) and \( y = -\frac{1}{3}x - 9 \)

We need both equations in slope-intercept form, \( y = mx + b \).

#### First equation: \( 5x + 15y = -8 \)

1. Solve for \( y \):
   \[
   15y = -5x - 8
   \]
   \[
   y = -\frac{5}{15}x - \frac{8}{15}
   \]
   \[
   y = -\frac{1}{3}x - \frac{8}{15}
   \]

So, the slope is \( m_1 = -\frac{1}{3} \).

#### Second equation: \( y = -\frac{1}{3}x - 9 \)

The slope is \( m_2 = -\frac{1}{3} \).

- \( m_1 = m_2 \), meaning the lines are **parallel** because they have the same slope.

**Conclusion**: The lines are **parallel**.

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### Final Answers:
(a) Neither parallel nor perpendicular.  
(b) Parallel.

Would you like further details or have any questions?

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### Related Questions:
1. What happens if the slopes are the same but the intercepts differ?
2. How do we find the slope from a standard form equation \( Ax + By = C \)?
3. What does it mean for two lines to be coincident?
4. Can perpendicular lines ever have the same slope?
5. How would the solution change if one line was vertical?

### Tip:
When comparing slopes, always ensure the equations are in slope-intercept form first!

Determine if the lines are parallel, perpendicular, or neither. Justify: 
a. y = -4x + 2 and y = 4x + 8
b. 5x + 15y = -8 and y = -1/3x - 9

Find out whether the given lines y = -4x + 2 and y = 4x + 8 are parallel, perpendicular, or neither. This problem covers key concepts such as the slope-intercept form, and tests your understanding of slopes and line relationships. Suitable for Grades 8-10 students.

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