Main content

### Course: Algebra basics > Unit 4

Lesson 5: Horizontal & vertical lines# Slope of a horizontal line

When two points have the same y-value, it means they lie on a horizontal line. The slope of such a line is 0, and you will also find this by using the slope formula. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- I know this question is "dumb" but can someone explain me why is Y/X and not X/Y? explain me like im 5 :)(29 votes)
- It is simply the way it has been defined. If some mathematician had decided hundreds of years ago that slope was run over rise, then we would all be taught that instead.

Of course, it probably also has a lot to do with how the coefficient is represented in slope-intercept form which also happens to be the most useful form for defining the line as a function.(20 votes)

- can someone explain to me how to get slope of parallel and perpendicular lines?(14 votes)
- Parallell lines have the same slope as the original line. Perpendicular lines are the negative reciprocal of the original lines.(6 votes)

- why is it called rise over run(11 votes)
- Simply put, the slope is called "rise over run" because to get from point A to point B, we rise (move vertically) a certain amount of units and then run (move horizontally) a certain amount of units.

Example: If we have a slope of 2 on the line (y = 2x + 1), we rise 2 units and run 1 unit to get from one point to another. Note: 2 can be thought of 2/1 -- they are the same thing, right? Start at the point (0, 1). The next point on the graph would be (1, 3). How did we get there? We first started by moving 2 units up in the y direction then got to our destination by moving 1 unit in the x direction.

Example: If we have a slope of 3/4 on the line (y = (3/4)x), we rise 3 units in the y direction and then 4 units in the x direction. Start at point (0,0). How would we find a second point on the graph? Well, we rise 3 units giving us a y value of (3) and run 4 units giving us an x value of (4); that gives us an ordered pair of (4,3).

What would be our third point? Well, from our second point at (4,3), we rise 3 units and run 4 units; that gives us a point at (8,6).(21 votes)

- Why when I do the questions/quiz/practice is the correct answer sometimes 'Undefined' and not 0?(5 votes)
- Great question!

It's something we definitely need to know, and we can calculate the answer as well!

★↔️**Horizontal lines**, (flat side to side),*always*have**Zero Slopes**.

When we calculate the slope…**Difference of y ÷ Difference of x**

∆y/∆x**the**, because the*Numerator will always equal zero***y value subtraction**on a Horizontal line**is always the same number minus itself**.

So…

Horizontal ↔️ Lines

the slope math: ∆y/∆x

will*always*= 0/∆x←zero on top**no matter what the Change in x is****, it will always divide**.*into*zero, zero times

Which is*why*a Horizontal line always has a Zero slope!

★↕️**Vertical lines**, (straight up and down),*always*have**Undefined Slopes**.

When we calculate the slope…**Change in y ÷ by Change in x**

∆y/∆x

the**x subtraction in the**, (a number minus itself), so it becomes a*Denominator*always equals zero**divide**situation,*by*zero*in arithmetic we learned*.**is Undefined**

So…

Vertical ↕️ Lines…

the slope math: ∆y/∆x*always*= ∆y/0←zero on bottombecause**It doesn't matter what the y-difference is****it's divided by**the x-difference (that is always**zero**), and**division by zero is**'**Undefined**'.

Which is*why*a Vertical line always has an Undefined slope!

★So if we forget which is which, we can calculate the slope and see for ourselves!

(≧▽≦) I hope that helps someone!(19 votes)

- anyone else find that the last 3 units were extremely hard to grasp compared to this?(12 votes)
- Hi, what is the difference between "0" and "undefined" in the quiz?(4 votes)
- A line with a slope of 0 is a horizontal line. A line with an undefined slope is a vertical line.(14 votes)

- At the practice of this, they say that the slope of a vertical line is always undefined.

why ?(4 votes)- Because there is no change in x which also mean there is no change in y, it won't be considered to have a slope. (It's a line but It has no slope)

In mathematical way: slope = y/0 = undefined

Also if you learn trigonometry, slope = tan a (opposite over adjacent)

vertical line is 90 degree, tan 90 = undefined

Edit:

My mistake, there's actually change in y, you can check the explanation in this answer comment.

Also, you can compare it with horizontal lines, it has a slope because there is a change of x but since it doesn't change y value, slope in horizontal line is 0

While in vertical, there is no change on x so we call it undefined.

This happen because slope is rate of change in line. In math, we usually define rate of change as: change of vertical over change of horizontal.(12 votes)

- To make an equation with a horizontal line of a slope of 0 would it be y= or x=?(6 votes)
- Is this question answerable: Find the slope of the line that goes through ordered pairs

(8,7) and (8,9). I know it is a vertical line but does that mean its impossible to calculate.(5 votes)- For the purpose of demonstration you can calculate it using rise over run: (7 - 9)/(8 - 8) = -2/0 <------hence a slope of infinity or undefined as mightygoose suggested because you can't divide by zero. So yes the question is answerable if you consider "undefined" to be an answer.(5 votes)

- Can you find slope using three values?(3 votes)
- You only need two points to find the slope. The slope between any two points on a line is always the same.(6 votes)

## Video transcript

Find the slope of the line
that goes through the ordered pairs 7, negative 1 and
negative 3, negative 1. Let me just do a
quick graph of these just so we can visualize
what they look like. So let me draw a
quick graph over here. So our first point
is 7, negative 1. So 1, 2, 3, 4, 5, 6, 7. This is the x-axis. 7, negative 1. So it's 7, negative 1
is right over there. 7, negative 1. This, of course, is the y-axis. And then the next point
is negative 3, negative 1. So we go back 3 in the
horizontal direction. Negative 3 for the y-coordinate
is still negative 1. So the line that connects these
two points will look like this. It will look like that. Now, they're asking us to find
the slope of the line that goes through the ordered pairs. Find the slope of this line. And just to give a little
bit of intuition here, slope is a measure of
a line's inclination. And the way that
it's defined-- slope is defined as rise over run, or
change in y over change in x, or sometimes you'll see it
defined as the variable m. And then they'll
define change in y as just being the
second y-coordinate minus the first
y-coordinate and then the change in x as the
second x-coordinate minus the first x-coordinate. These are all different
variations in slope, but hopefully you'll
appreciate that these are measuring inclination. If I rise a ton when
I run a little bit, if I move a little bit in the x
direction, and I rise a bunch, then I have a very steep line. I have a very steep
upward-sloping line. If I don't change at
all when I run a bit, then I have a very low slope. And that's actually
what's happening here. I'm going from-- you
could either view this as the starting
point or view this as the starting point. But let's view this
as the starting point. So this negative 3, 1. If I go from negative 3,
negative 1 to 7, negative 1, I'm running a good bit. I'm going from negative 3. My x value is negative 3 here,
and it goes all the way to 7. So my change in x here is 10. To go from negative 3 to 7,
I changed my x value by 10. But what's my change in y? Well, my y value
here is negative 1, and my y value over here
is still negative 1. So my change in y is a 0. My change in y is going to be 0. My y value does not
change no matter how much I change my x value. So the slope here
is going to be-- when we run 10,
what was our rise? How much did we change in y? Well, we didn't rise at all. We didn't go up or down. So the slope here is 0. Or another way to think about
is this line has no inclination. It's a completely flat-- it's
a completely horizontal line. So this should make sense. This is a 0. The slope here is 0. And just to make
sure that this gels with all of these other
formulas that you might know-- but I want to make
it very clear. These are all just
telling you rise over run or change in y over change in
x, a way to measure inclination. But let's just
apply them just so, hopefully, it all
makes sense to you. So we could also say slope is
change in y over change in x. If we take this to be
our start and if we take this to be our
end point, then we would call this over here x1. And then this is over here. This is y1. And then we would
call this x2 and we would call this y2, if
this is our start point and that is our end point. And so the slope here, the
change in y, y2 minus y1. So it's negative 1
minus negative 1, all of that over x2, negative
3, minus x1, minus 7. So the numerator, negative
1 minus negative 1, that's the same thing
as negative 1 plus 1. And our denominator
is negative 3 minus 7, which is negative 10. So once again, negative 1
plus 1 is 0 over negative 10. And this is still going to be 0. And the only reason why
we got a negative 10 here and a positive 10
there is because we swapped the starting
and the ending point. In this example right over here,
we took this as the start point and made this coordinate
over here as the end point. Over here, we
swapped them around. 7, negative 1 was
our start point, and negative 3, negative
1 is our end point. So if we start over
here, our change in x is going to be negative 10. But our change in y is
still going to be 0. So regardless of how you do it,
the slope of this line is 0. It's a horizontal line.