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!