The problem provides six linear equations in the form $y = mx + c$. We need to identify two equations that are parallel and two equations that are perpendicular.

1. Problem Description

The problem provides six linear equations in the form $y = mx + c$. We need to identify two equations that are parallel and two equations that are perpendicular.

2. Solution Steps

Parallel Lines: Two lines are parallel if they have the same slope ($m$) but different y-intercepts ($c$).

Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -

1. If line 1 has slope $m_1$ and line 2 has slope $m_2$, then $m_1 * m_2 = -1$. This means that the slope of one line is the negative reciprocal of the slope of the other line.

The given equations are:

i) $y = 3x + 4$

ii) $y = 4x + 2$

iii) $y = -4x + 4$

iv) $y = -\frac{1}{4}x + 7$

v) $y = \frac{1}{3}x - 2$

vi) $y = 3x - 5$

Parallel Lines:

Comparing the slopes of the equations, we can see that equations i) and vi) have the same slope ($m = 3$). Since their y-intercepts are different ($4$ and $-5$ respectively), the lines are parallel.

Perpendicular Lines:

We need to find two lines whose slopes multiply to -

1. Let's examine equation ii) $y = 4x + 2$, which has a slope of

4. Its negative reciprocal is $-\frac{1}{4}$. Equation iv) has a slope of $-\frac{1}{4}$.

Thus, equation ii) and equation iv) are perpendicular.

3. Final Answer

a) Parallel: i) and vi)

b) Perpendicular: ii) and iv)