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## High school geometry

### Course: High school geometry > Unit 6

Lesson 5: Equations of parallel & 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
- Analytic geometry FAQ

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# Parallel lines from equation

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?

- I need help with finding the equation of a line parallel to another line. Say the first line has a slope of 2/5 and passes through (3, -5). How would I get the equation of a line parallel to that? My math book ain't helpin'.(17 votes)
- Since you know the slope of the first line and one of the points it passes through, write the equation of the first line in point-slope form (y-y1=m(x-x1)) by substituting the slope, x-coordinate of the point, and y-coordinate of the point for m, x1, and y1, respectively. This will give you y-(-5)=2/5(x-3), which you can simplify to y+5=2/5(x-3).

Then, convert this equation to slope-intercept form by isolating y. Using the Distributive Property, 2/5*x=2/5x and 2/5*3=6/5. Using the Subtraction Property of Equality, 6/5-5=-19/5. So the equation in slope-intercept form is y=2/5x-19/5.

Parallel lines have equal slopes, so the second line will also have a slope of 2/5. Therefore, to get the equation of a line parallel to the first line, all you have to do is change the y-intercept, -19/5. For example, the lines y=2/5x, y=2/5x+2, and y=2/5x-4 are all parallel to y=2/5x-19/5, just to name a few.(1 vote)

- How to prove lines parallel(4 votes)
- You have to find the slopes of the two lines, if they are the same, then the lines are parallel unless they are the exact same line.(4 votes)

- does it matter if one of the slopes is negative and the other is positive? are they still parallel or not?(2 votes)
- Parallel lines will always have the same slope, so there cannot be one positive and one negative, these lines will always intersect somewhere. Perpendicular lines almost always have opposite signs, so they could possibly be perpendicular.(3 votes)

- at around2:54Sal says you increase by one. I didn't see that in the equation anywhere, how did he get that?(2 votes)
- He just picked a random number to increase by. He could have increased by 1,2,3,4,5.....

He would just have to make sure that whatever number he**did**pick, he would multiply it by two to know how much to go up.

Does that make sense? Please tell me if it doesn't!(3 votes)

- What do you do if it tells you y=x? Like for example;

y=x

x+Y=-2?(2 votes)- If you subtract x on both sides, you will get y=-x+2. This shows that the slopes are opposite reciprocals (1/1 and -1/1) and thus lines are perpendicular. If you just have a variable, the coefficient is an invisible 1 (or in slope terms 1/1).(3 votes)

- what if you have y=3x+4 and it passes through (-2,3)(2 votes)
- I'm assuming you want a line parallel to y = 3x + 4. All this means is that the slope will be 3 and the y intercept will not be 4. or in otherwords y = mx+b, m = 3 and b doesn't equal 4

Now, as long as you have a slope and point you can find the equation of the line. And you do, you want a slope of 3 and a point (-2,3). To use this you use point slope form. y - y1 = m(x - x1) where (x1, y1) is the point you want. and m is slope. So just fill in and solve.

y - y1 = m(x - x1) with m = 3 and (x1, y1) = (-2,3)

y - 3 = 3(x - -2)

If you prefer KC's way just start with y = 3x + b then b = y - 3x where you plug in (-2, 3) for x and y, or in other words:

b = y - 3x where x = -2 and y = 3. Then again just plug in and solve. then plug b into y = 3x + b

You get the same answer either way.(3 votes)

- I have a question, please answer it. Y = -4/3x + 6. Is this Perpendicular?(1 vote)
- perpendicular to what other line? Two lines are perpendicular if they have opposite reciprocal slopes, so any line with a slope of 3/4 would be perpendicular to this line such as y = 3/4 x(4 votes)

- I don't understand every thing and i want to boost my grade(2 votes)
- i dont understand the concept at1:32(1 vote)
- Slope intercept form is:

change in y

------------

change in x

slope intercept form tells us how much*y_ changes for every _x**Line A*has a slope of 2, so basically that means for every**2**that*y_ changes by, _x*changes by**1**(2 votes)

- but at0:57he just told us the answer right?(1 vote)

## Video transcript

We are asked which of these
lines are parallel. So parallel lines are lines that
have the same slope, and they're different lines, so they
never, ever intersect. So we need to look for different
lines that have the exact same slope. And lucky for us, all of these
lines are in y equals mx plus b or slope-intercept form, so
you can really just look at these lines and figure
out their slope. The slope for line A,
m is equal to 2. We see it right over there. For line B, our slope is equal
to 3, so these two guys are not parallel. I'll graph it in a second
and you'll see that. And then finally, for line C--
I'll do it in purple-- the slope is 2. So m is equal to 2. I don't know if that purple
is too dark for you. So line C and line A have the
same slope, but they're different lines, they have
different y-intercepts, so they're going to be parallel. And to see that, let's actually
graph all of these characters. So line A, our y-intercept
is negative 6. So the point 0, 1,
2, 3, 4, 5, 6. And our slope is 2. So if we move 1 in the positive
x direction, we go up 2 in the positive y direction. One in x, up 2 in y, if we
go to in x, we're going to go up 4 in y. And I can just do up 2, then
we're going to go 2, 4, and you're going to see it's all on
the same line, so line A is going to look something like--
do my best to draw it as straight as possible. Line A-- I can do a better
version than that-- line A is going to look like-- well,
that's about just as good as what I just drew--
that is line A. Now let's do line B. Line B, the y-intercept
is negative 6. 0, negative 6. So it has the same y-intercept,
but its slope is 3, so if x goes up by 1,
y will go up by 3. So x goes up by 1,
y goes up by 3. If x goes up by 2, y is
going to go up by 6. 2, 4, 6. So this line is going to look
something like this. Trying my best to connect
the dots. It has a steeper slope, and you
see that when x increases, this blue line increases by
more in the y direction. So that is line B-- and notice,
they do intersect, there's definitely not
two parallel lines. And then finally, let's
look at line C. The y-intercept is 5. So 0, 1, 2, 3, 4, 5. The point 0, 5, its
y-intercept. And its slope is 2. So you increase by 1 in the x
direction, you're going to go up by 2 in the y direction. If you decrease by 1,
you're going to go down 2 in the y direction. If you increase by, well, you're
going to go to that point, you're going to have
a bunch of these points. And then if I were to graph the
line-- let me do it one more time-- if I were to
decrease by two, I'm going to have to go down by 4, right? Negative 4 over negative
2 still a slope of 2, so 1, 2, 3, 4. And I can do that one more time,
get right over there. And then you'll see the line. The line will look like that,
it will look just like that. And notice that line C and
line A never intersect. They have the exact
same slope. Different y-intercepts, same
slope, so they're increasing at the exact same rate, but
they're never going to intersect each other. So line A and line
C are parallel.