Question 4c0bb

Jan 31, 2018

When given two lines in slope-intercept form:
$y = {m}_{1} x + {b}_{1}$
$y = {m}_{2} x + {b}_{2}$
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:

$y = 2 x + 5$ and $y = 2 x + 6$ 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 ${C}_{1} \ne {C}_{2}$.

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

${m}_{1} {m}_{2} = - 1$

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:

$y = 2 x + 5$ and $y = - \frac{1}{2} x + 6$ 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 ${C}_{2}$

Feb 1, 2018

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 $- 1$, $\left({m}_{1} {m}_{2} = - 1\right)$.

Explanation:

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

Slope-Intercept Form

$y = m x + b$,

where:

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

Substitute $0$ for $x$.

$y = m \left(0\right) = b$

$y = b$

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

Parallel Lines

Example: Are the following lines parallel?

$y = 3 x + \textcolor{b l u e}{4}$

$y - \textcolor{b l u e}{4} = 3 x$ $\leftarrow$ Point-slope form.

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

$y = 3 x + \textcolor{b l u e}{4}$

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

Perpendicular Lines

Example: Are the following lines perpendicular?

$y = \textcolor{b l u e}{5} x + 8$

$y = \textcolor{g r e e n}{- \frac{1}{5}} x + 8$

Multiply the slopes.

${m}_{1} {m}_{2} = {\textcolor{red}{\cancel{\textcolor{b l u e}{5}}}}^{1} \times \left(\textcolor{g r e e n}{- \frac{1}{\textcolor{red}{\cancel{\textcolor{g r e e n}{5}}}} ^ \textcolor{b l a c k}{1}}\right) = - 1$

Therefore the lines are perpendicular.

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

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