Question

b. what are these lines are what compared to each other?
y = 5/7x + 2
y = 5/7x - 30

c. create a line that is perpendicular to
y = -4x + 8

notes:

Explanation:

Part B

Step1: Recall slope - intercept form

The slope - intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
For the line \(y=\frac{5}{7}x + 2\), the slope \(m_1=\frac{5}{7}\) and for the line \(y=\frac{5}{7}x-30\), the slope \(m_2 = \frac{5}{7}\).

Step2: Determine the relationship between lines

Two lines are parallel if their slopes are equal (\(m_1=m_2\)) and they have different y - intercepts (to ensure they are not the same line). Here, \(m_1 = m_2=\frac{5}{7}\) and the y - intercepts are \(2\) and \(- 30\) (which are different). So the two lines are parallel.

Step1: Recall the slope of a perpendicular line

If a line has a slope \(m\), then the slope of a line perpendicular to it, \(m_{\perp}\), is the negative reciprocal of \(m\), i.e., \(m_{\perp}=-\frac{1}{m}\) (when \(m
eq0\)).
For the line \(y=-4x + 8\), the slope \(m=-4\).

Step2: Calculate the slope of the perpendicular line

The negative reciprocal of \(-4\) is \(\frac{1}{4}\) (since \(-\frac{1}{-4}=\frac{1}{4}\)).

Step3: Write the equation of the perpendicular line

We can use the slope - intercept form \(y = m_{\perp}x + b\). We can choose any value for \(b\) (let's choose \(b = 0\) for simplicity). So an equation of a line perpendicular to \(y=-4x + 8\) is \(y=\frac{1}{4}x\) (we could also choose other values for \(b\), for example, if \(b = 5\), the line would be \(y=\frac{1}{4}x+5\), etc.).

Answer:

The two lines \(y=\frac{5}{7}x + 2\) and \(y=\frac{5}{7}x-30\) are parallel to each other.

Part C