Question

Question #4c0bb

Answer 1

When given two lines in slope-intercept form:
`y=m_1x+b_1`
`y = m_2x+b_2`
Only the slopes play a role in determining whether the lines are parallel or perpendicular.

Explanation:

Two lines are parallel, if and only if their slopes are equal:

`m_1 = m_2`

When the two slopes are equal, it can be said that the two lines belong to a family of lines that are parallel.

For example, we know that the two lines:

`y = 2x+5` and `y = 2x+6` are parallel, because their slopes are equal to 2.

There is are two special cases for parallel lines.

1 Vertical lines are parallel; their slope is undefined and they have no y intercept. They have the form:

`x = C_1; C_1 in RR`
`x = C_2; C_2 in RR`

Any two lines of this form parallel.

2 Horizontal lines are parallel; their slope is 0. They will have different y-intercepts. They have the form:

`y = C_1; C_1 in RR`
`y = C_2; C_2 in RR`

Any two lines of this form are parallel.

NOTE: In both cases `C_1!=C_2`.

Two lines are perpendicular, if and only if the product of their slopes is equal to -1:

`m_1m_2= -1`

When the product of the two slopes is equal to -1, it can be said that the two lines belong to a family of lines that are perpendicular.

For example, we know that the two lines:

`y = 2x+5` and `y = -1/2x+6` are perpendicular, because the product of their slopes is equal to -1.

There is a special case for perpendicular lines where one is a horizontal line and the other is a vertical line:

`y = C_1; C_1 in RR`
`x = C_2; C_2 in RR`

Any two lines of these forms are perpendicular.

NOTE: `C_1` may or may not equal `C_2`

Answer 2

Yes for parallel lines. Lines that are parallel cannot have the same y-intercept. If they do, then they are the same line. Two lines are perpendicular if the product of their slopes equal `-1`, `(m_1m_2=-1)`.

Explanation:

The y-intercept is the value of `y` when `x=0`.

Slope-Intercept Form

`y=mx+b`,

where:

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

Substitute `0` for `x`.

`y=m(0)=b`

`y=b`

So, if the y-intercept `(b)`, is the same for both lines, then they are the same line.

Parallel Lines

Example: Are the following lines parallel?

`y=3x+color(blue)4`

`y-color(blue)4=3x` `larr` Point-slope form.

Solve for `y` for the point-slope form.

`y=3x+color(blue)4`

They are the same line, so they are not parallel.

Perpendicular Lines

Example: Are the following lines perpendicular?

`y=color(blue)5x+8`

`y=color(green)(-1/5)x+8`

Multiply the slopes.

`m_1m_2=color(red)cancel(color(blue)(5))^1xx(color(green)(-1/color(red)cancel(color(green)(5))^color(black)1))=-1`

Therefore the lines are perpendicular.

Notice on the graph that the point of intersection is the y-intercept:
`b=(0,8)`

graph{(y-5x-8)(y+1/5x-8)=0 [-17.35, 14.67, -2.88, 13.14]}