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### Course: Algebra (all content) > Unit 3

Lesson 4: Horizontal & vertical lines# Horizontal & vertical lines

Worked examples identifying the equations and slope of horizontal and vertical lines.

## Want to join the conversation?

- Why is an equation in which you only have the value of y considered to have a slope of 0 (for example the equation at2:00) but an equation in which you only have the slope of x (for example the equation at3:05) considered to be undefined?(39 votes)
- Great question! The slope is the change in y over the change in x. In the horizontal line at2:00, there is zero change in y, which means that the slope is 0 divided by the change in x. 0 divided by any number is 0, giving you a slope of 0.

For the example at3:05, there is 0 change in x which means that the slope is the change in y over 0. Since you can't divide by 0, the slope is undefined.

If you don't understand why any number divided by 0 is undefined, I think this video explains it pretty well: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:foundation-algebra/x2f8bb11595b61c86:division-zero/v/why-dividing-by-zero-is-undefined

Hope this helps! Great question and have a nice day!(30 votes)

- I should really be paying attention to the video instead of scrolling through comments.(14 votes)
- So if the line is vertical the slope is undefined and if the line is horizontal the slope is 0. Am I understanding it right?(13 votes)
- Dividing by 0 is undefined, this is why a vertical line has an undefined slope.

If this doesn't make sense, let me give you an example. let's say our change in y = 10

and for a vertical line, the change in x = 0 , the slope then equals to change in y/change in x. So 10/0, again, dividing by zero is undefined(3 votes)

- Why does a horizontal line have a slope of O, and a vertical line has an undefined slope?(0 votes)
- Remember, the slope of a line is "change in Y / change in X". You need to think of the slope as a fraction. The numerator tells you the change in Y -- how fast the Y coordinate is changing (how fast the line is going up/down). The denominator tells you the change in X -- how fast the X coordinate is changing (how fast the line is moving left/right).

Horizontal lines do not go up/down. They just move left to right. This means the change in Y = 0, while the change in X = 1. 0/1 = 0 as a slope.

Vertical lines go up/down, but they never go left or right. This means the change in Y = 1, while the change in X = 0. 1/0 = undefined.

Hope this helps.(20 votes)

- ehh so if we have the equation of a vertical line, like X=4...it has the possibility for Y to have any value.. but why isn't the Y represented in the equation at all? Couldn't you say x=4 + y(any real numbers) ?(3 votes)
- Think of it this way... the reason "y" is not in the equation is because its coefficient = 0.

x + 0y = 4

You can't make it into x = 4 + y because it would no long be a vertical line.

Hope this helps.(10 votes)

- I do not get how a vertical line is undefined...(2 votes)
- Remember, slope is (Change in Y)/(Change in X). The slope of a vertical line is undefined because the Change in X = 0 and dividing by 0 is undefined.

Hope this helps.(12 votes)

- At2:11, isn't it a change of -4 because y=-4?(5 votes)
- y=-4 is like y= 0x -4. In order for the x to be 'gone' you would need a zero at its coefficient. since the coefficient of the x is the slope then the slope is zero.(5 votes)

- What's the difference between "0" and "undefined" as slope?(3 votes)
- Since the slope intercept general equation is y = mx + b, if you substitute 0 in for m, then 0x = 0 which disappears, so y = b. Zero slope gives a horizontal line that crosses the x axis at a given y value.

Then, if we tried to substitute an "undefined" value for m, we really could not, but lets just imagine that we try. 1/0 is the simplest undefined number. So y = 1/0 x + b

and once again imagining that I could multiply by 0 to get rid of it on the bottom (but 0/0 is still undefined), we would have 0y = 1x +ob, or just x=0. Thus, this would be a vertical line, but x is not limited to just 0, it could be any number. This vertical line will cross the y axis at a given x value. This would mean that it would also is not a function.(7 votes)

- What does it mean when Sal said "It doesn't matter what the change in x is and that the change in y will always equal zero" at2:09. Why would it equal zero if slope is rise over run or y/x and y would always equal -4 because it runs in a horizontal line and is consistent? If x could be any number possible on the x axis, why does the slope equal zero?(2 votes)
- Yes, Y always = -4. This means the change in Y = y2-y1 = -4-(-4) = -4+4 = 0. The change in Y always = 0, because Y is not changing. The line has no up/down movement, so the slope = 0. This is what makes the line a horizontal line.

Hope this helps.(7 votes)

## Video transcript

- [Instructor] What is the equation of the horizontal line through the point negative four comma six? So let's just visualize this. Once you get the hang of it, you might not have to draw a graph, but for explanatory
purposes, it might be useful. So negative four comma six, that's going to be in the second quadrant. So if this is my x-axis, That is my y-axis. I'm going to go negative
four in the x direction. So one, two, three, four. Negative four. And then one, two, three, four, five, six, in the y direction. So the point that we care about is going to be right over there. Negative four comma six. And what is the equation
of the horizontal line? It is a horizontal line. So it's just going to go straight left, right like this. That is what the line
would actually look like. So what is that equation? Well, for any x, y is going to be equal to six. This is the equation, y is equal to six. Doesn't matter what x you input here, you're gonna get y equals six. It just stays constant right over there. So the equation is y is equal to six. Let's do another one of these. So here we are asked what is the slope of the line y is equal to negative four? So let's visualize it and then in the future, you might not have to draw it like this. But let's just draw our axis again. X-axis y-axis and the slope of line
y equals negative four. So for whatever x you have, y is going to be negative four. Let's say that's negative
four right over there. And so, the line is y the line is y equals negative four. So I can draw it like this. So what's the slope of that? Well, slope is change in
y for given change in x. And here, no matter what I change my x, y doesn't change. It stays at negative four. My change in y over change in x. Doesn't matter what my change in x is. My change in y is always going to be zero. It's constant. So the slope here is
going to be equal to zero. Y doesn't change, no matter how much you change x. Let's do another one of these. This is fun. So now they are asking us, what is the slope of the line x equals negative three? Let me graph that one. So, I'm just going to draw my axis real fast. X-axis y-axis X is equal to negative three. So negative one, negative two, negative three. And so, this line is going to look let me, it's going to look like this. No matter what y or you can say no matter what y is. X is going to be equal to negative three. So, it would look like this. X is equal to negative three. So what's the slope here? Well, it's undefined. A vertical line has an undefined slope. Remember, you want to do
what's your change in y or change in x. Change in y or change in x. Well, you can think about what's the slope as you approach this but once again, that could be, some people would say, maybe it's infinite, maybe
it's negative infinity. But that's why it's undefined. A vertical line is going
to have an undefined slope. So we'll go with undefined. Let's do one more. What is the equation of the vertical line through negative five comma negative two? So, let me do this without even drawing it. I'll draw it right after that. So, if we're talking about a vertical. If we're talking about a vertical line, that means that x doesn't change. X doesn't change. If we were talking
about a horizontal line, then we'd say y doesn't change. So if x doesn't change, that means that x is just going to be equal
to some constant value. Well, if it contains
the points negative five comma negative two, so if it has a point where
x is equal to negative five and if x never changes,
it's a vertical line, well that means its equation has to be x is equal to negative five. And we can draw that out, if it helps. So let me draw that out. So, I need to make sure
that's a straight line. Okay, so we have x, and we have y, so we have the point negative
five comma negative two. So negative one, two, three, four, five, negative one, two so we want to go have a vertical line that goes through that point. So a vertical line, well that just goes straight up and down. So it's just going to look like this. And so notice, x never changes. No matter what y is, x is equal to negative five. This has an undefined slope. It is a vertical line. Its equation is x is
equal to negative five.