Main content

## Geometry (all content)

### Course: Geometry (all content) > Unit 15

Lesson 5: Equations of parallel and perpendicular lines- Parallel lines from equation
- Parallel lines from equation (example 2)
- Parallel lines from equation (example 3)
- Perpendicular lines from equation
- Parallel & perpendicular lines from equation
- Writing equations of perpendicular lines
- Writing equations of perpendicular lines (example 2)
- Write equations of parallel & perpendicular lines
- Proof: parallel lines have the same slope
- Proof: perpendicular lines have opposite reciprocal slopes

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Parallel lines from equation (example 3)

Sal determines which pairs out of a few given linear equations are parallel. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- When two lines have the same slope, but they also have the same y-intercept (they lie on each other), are they considered parallel?(53 votes)
- Excellent question,

If two lines had the same slope and same y-intercept, they would not be considered parallel lines.

They would be considered as coincident lines--lines that lie directly on top of each other.(69 votes)

- At1:05for the slope formula of y=6 i thought the slope is 6 and the y intercept is 0? Is that right? If not, why isn't it?(5 votes)
- If the slope was 6, there would be a x after the 6.(11 votes)

- (5, -5), parallel to y = - 3\5 x + 2(7 votes)
- Using y=mx+b

if it is parallel, means it has the same slope.

So your slope m = -3/5

the coordinate (x,y) is (5,-5)

So your x=5 and y=-5

Now you can substitute them into y=mx+b and solve for b.(6 votes)

- at0:29why do you divde them all by 2??(2 votes)
- Line A equation is given as 2y = 12x + 10

We use slope of the line to find out if lines are parallel. Slope is read easily from the slope-intercept form, which looks like this:

y = ax + b (where a is the slope)

So how can we easily turn 2y = 12x + 10 into something that starts with "y = "?

.

.

.

We divide both sides of the equation by 2 :)(7 votes)

- Are lines that are negative reciprocals perpendicular?(2 votes)
- Yes, but you will probably often see different wording. The way you might hear it is that perpendicular lines have opposite reciprocal slopes. You should have the word slope somewhere in your question. The idea is that if it is positive, the perpendicular line would have a negative slope, but if the line has a negative slope, then the perpendicular would have a positive slope. If the slope of a line is -3, then the perpendicular line has a slope of 1/3.(6 votes)

- hello. i don't know how to solve this problem. i have to find the equation:

intersects with y=2x-1 on the y-axis and is parallel to y=-x

Thank you!(3 votes)- y = 2x - 1 implies that the y-intercept for the equation you are looking for is -1.

y = -x implies that the slope of the equation you are looking for is -1.

Plugging in these two pieces of information into the slope intercept form (y = mx + b) results in:

y = -x - 1(6 votes)

- If I have 2 slopes for example 150\120 and my second slope 125/100 are they opposite reciprocal slope?(3 votes)
- What if for Line B x=6 what would we need to do then?(3 votes)
- How would I write an equation with parallel lines? An example is: write an equation through (9,-1) and parallel yo y=-5x+7.(2 votes)
- Your given line y = -5x +7, is in slope intercept form, so you can see the slope is -5.

Use the point slope form to quickly write an equation. Because the line you want is parallel, you know it has the same slope as your given line.

So the parallel line will also have slope -5. It goes thru (9.-1);

Point slope form is y - y1 = m (x - x1); here y1 = -1 and x1= 9 and m=-5

y - -1 = -5(x -9)

y+1 = -5(x-9)

With a little algebra you can change this to slope intercept form if you need that.(3 votes)

- When two lines have the same slope, but they also have the same y-intercept (they lie on each other), are they considered parallel? And how(2 votes)
- The better math word is coincidental lines. Parallel implies different intercepts.(2 votes)

## Video transcript

We're asked which of these
lines are parallel. So they give us three equations
of three different lines and if they're parallel,
then they have to have the same slope. So all we have to do over here
is figure out the slopes of each of these lines, and
if any of them are equal, they're parallel. So let's do line A. Line A, it's 2y is equal
to 12x plus 10. We're almost in slope-intercept
form, we can just divide both sides of
this equation by 2. We get y is equal to 6x--
right, 12 divided by 2 -- 6x plus 5. So our slope in this case, we
have it in slope-intercept form, our slope in this
case is equal to 6. Let's try line B. Line B is y is equal to six. You might say this hey, this is
a bizarre character, how do I get this into slope-intercept form, where's the x? And my answer to you is
that it already is in slope-intercept form. I could just rewrite it as
y is equal to 0x plus 6. The x term is being multiplied
by 0 because the slope here is 0. y is going to be equal
to six no matter how much you change x. Change in y is always
going to be 0, it's always going to be 6. So here, our slope is 0, so
these two lines are definitely not parallel, they have
different slopes. So let's try line C. Line C-- I'll do it down here. Line C, so it's y minus 2 is
equal to 6 times x plus 2. And this is actually in
point-slope form, where the point x is equal to negative
2, y is equal to 2. So the point negative 2, 2, is
being represented here because you're subtracting the points. And the slope is 6, so we
already know that the slope is equal to 6. And sometimes people are
more comfortable with slope-intercept form, so let's
put it in slope-intercept form just to confirm that if we put
it in this form, the slope will still be equal to 6. So if we distribute the 6, we
get y minus 2 is equal to 6 times x, 6x, plus
6 times 2 is 12. And if you add this 2 -- if you
add 2 to both sides of the equation, you get y-- because
these guys cancel out-- is equal to 6x plus 14. So you see, once again,
the slope is 6. So line A and line C have the
same the slope, so line A and line C are parallel. And they're different lines. If they had the same
y-intercept, then they would just be the same line.