Question

X + 7y = -3 and y = 7x + 25. Are these parallel, perpendicular, or neither.

Answer 1

In order to know if those lines are parallel, perpendicular or neither we need to compare the slopes.

When two lines are parallel they have the same slope, when they are perpendicular their slopes are negative reciprocal, which means:

`\begin{gathered} m_1=-(1)/(m_2) \\ \text{Then} \\ m_1\cdot m_2=-1 \end{gathered}`

Now, we need to arrange both equations into the slope-intercept form y=mx+b

Where m is the slope and b is the y-intercept.

The second line is already in slope-intercept form:

`y=7x+25`

Thus, its slope is 7.

The first line in slope-intercept form is:

`\begin{gathered} x+7y=-3 \\ 7y=-3-x \\ y=(-3-x)/(7) \\ y=(-3)/(7)-(x)/(7) \\ \text{ By reordering terms} \\ y=-(x)/(7)-(3)/(7) \end{gathered}`

Then it slope is -1/7.

Their slopes are not the same, then they aren't parallel, but let's check if they are perpendicular:

`\begin{gathered} m1\cdot m2=-1 \\ -(1)/(7)\cdot7=-1 \\ -1=-1 \end{gathered}`

They are negative reciprocal, then they are perpendicular lines.