Let's solve the question by analyzing each pair of equations to determine whether the lines are parallel, perpendicular, or neither.

General Approach:

1. Parallel lines: Two lines are parallel if their slopes are the same.
2. Perpendicular lines: Two lines are perpendicular if the product of their slopes is $-1$.
3. Neither: If neither condition is met.

We'll convert each equation to slope-intercept form ($y = mx + b$), where $m$ is the slope.

a) $y = 2x + 3$ and $2y = x + 10$

• First equation: Already in slope-intercept form. Slope $m_1 = 2$.
• Second equation: Solve for $y$.

$2y = x + 10 \quad \Rightarrow \quad y = \frac{1}{2}x + 5$

Slope $m_2 = \frac{1}{2}$.

• Since $m_1 = 2$ and $m_2 = \frac{1}{2}$, the lines are neither parallel nor perpendicular.

b) $y = 5x + 2$ and $y + x = 5$

• First equation: Already in slope-intercept form. Slope $m_1 = 5$.
• Second equation: Solve for $y$.

$y + x = 5 \quad \Rightarrow \quad y = -x + 5$

Slope $m_2 = -1$.

• Since $m_1 = 5$ and $m_2 = -1$, the lines are neither parallel nor perpendicular.

c) $y = 4x - 1$ and $4y + 16x = 5$

• First equation: Already in slope-intercept form. Slope $m_1 = 4$.
• Second equation: Solve for $y$.

$4y = -16x + 5 \quad \Rightarrow \quad y = -4x + \frac{5}{4}$

Slope $m_2 = -4$.

• Since $m_1 = 4$ and $m_2 = -4$, the product of the slopes is $(4)(-4) = -16$, which is not equal to $-1$. So the lines are neither parallel nor perpendicular.

d) $y = -4x - 1$ and $4y + 16x = 5$

• First equation: Already in slope-intercept form. Slope $m_1 = -4$.

• Second equation: From part c), we know the slope is $m_2 = -4$.

• Since $m_1 = -4$ and $m_2 = -4$, the slopes are equal, so the lines are parallel.

Final Answers:

a) Neither
b) Neither
c) Neither
d) Parallel

Would you like a detailed explanation of any specific part or need further assistance?

Here are 5 questions to think about:

1. How do you find the slope of a line from a given equation?
2. What are the conditions for two lines to be parallel?
3. What does it mean for two lines to be perpendicular?
4. Can two lines with the same slope but different intercepts ever intersect?
5. What is the significance of the slope-intercept form of a line?

Tip: Always convert equations to slope-intercept form to easily compare slopes and determine relationships between lines.