*Something To Think About* I think that when x_2 = x_1 then the slope will become undefined because x_2 - x_1 equals zero. Therefore when you divide y_2 - y_1 it won't be possible. *Example* (5,10) (5,15) x_1 = 5 x_2 = 5 y_1 =10 y_2 =15 5 - 5= 0 15-10=5 5/0= *Undefined*

Yes, you are correct. The slope of any line through two different points with the same x-coordinate (that is, a vertical line) is always undefined, for the reason you stated.

Why is the slope formula y/x? Why not y-x or y+x? Thanks to anyone who answers. Jack.

Slope is something that is also referred to as the rate of change. For example, if you had a savings account that you deposited no money into initially but you deposit 20$ weekly, your rate of change, or slope for this problem would be 20. This is because your x-value in this situation would be the number of weeks passed since you have created your bank account, and the y-value is how much money you have deposited into your account, fully. Since you are looking at the rate of change between the weeks, you divide the change in y per week, 20, by 1 for the number of weeks. I hope this somewhat answers your question.

If I Get The Right Answer Then Why Do I Have To Simplify?

Simplifying just makes it easier to read/understand. It makes it more "simple." Although both are equal, it is just easier to work with if it's simplified afterwards.

For the last one, if x_1 equals x_2 it is undefined, this is because, from the other videos, it was said that if the two points have the same x when drawing the line, it will be straight up and down, with no slope, but those are called undefined as there really is an undefined slope to it

Yes you are correct that the slope is undefined if x_1 = x_2. Good job!

bro im honestly so gone

Slope is basically the change in the y direction divided by the change in the x direction. If you don't know graphs, you might want to learn that first.

in the formula mx+b=y, I understand that "m" is the slope and "b" is the y-intercept, but what is x and y?

when you graph the line, mx+b=y and fill in the slope and y-intercept, the x and y represent points that are on the line that you graphed. For example, if the equation was 5x+10=y, you could create pairs of (x,y) coordinates by plugging in numbers for x and y. In this case, if x was 5, y would be 35 or vice versa. Based on this, you could say that (5,35) is a point on the line, 5x+10=y.

In the very last part, why is the formula for slope delta y/delta x instead of delta x/ delta y?

This is because the equation that describes a line is y=mx+c. If we have the y and x values (as in the coordinates), and c is constant for both points (which if it is two point on one line, we know it is) than we can solve for m with algebra. If we have two coordinates on a line (x1,y1 =1,2) and (x2, y2 =3,6) we can solve for m as follows. (x2,y2) 6=m3+c - (x1,y1) 2=m1+c 1st step: c-c =0 we are left with 6=m3 - 2=m1 The first equation minus the second = 4=2m But we want the slope (m) on one side so we can solve for M. 4/2=m 2=m which is your slope What you have done here is take y2 from y1 on the left, x2 from x1 on the right, then divided by x to get m on its own. We can do this in one step instead to get the slope by the equation (y2-y1)/(x2-x1)=m That is why you divide by x rather than Y.

Main content

8th grade (Eureka Math/EngageNY)

Course: 8th grade (Eureka Math/EngageNY) > Unit 4

Lesson 3: Topic C: Slope and equations of lines

Slope formula

CCSS.Math:

8.F.B.4

Google Classroom

Learn how to write the slope formula from scratch and how to apply it to find the slope of a line from two points.

It's kind of annoying to have to draw a graph every time we want to find the slope of a line, isn't it?

We can avoid this by writing a general formula for slope. Before we start, let's remember how slope is defined:

start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction

Let's draw a line through two general points left parenthesis, start color #1fab54, x, start subscript, 1, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 1, end subscript, end color #e07d10, right parenthesis and left parenthesis, start color #1fab54, x, start subscript, 2, end subscript, end color #1fab54, comma, start color #e07d10, y, start subscript, 2, end subscript, end color #e07d10, right parenthesis.

An expression for start color #1fab54, start text, c, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54 is start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54:

[Why does this expression work?]

Similarly, an expression for start color #e07d10, start text, c, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10 is start color #e07d10, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10:

Now we can write a general formula for slope:

start text, S, l, o, p, e, end text, equals, start fraction, start color #e07d10, start text, C, h, a, n, g, e, space, i, n, space, y, end text, end color #e07d10, divided by, start color #1fab54, start text, C, h, a, n, g, e, space, i, n, space, x, end text, end color #1fab54, end fraction, equals, start fraction, start color #e07d10, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, end color #e07d10, divided by, start color #1fab54, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end color #1fab54, end fraction

That's it! We did it!

Using the slope formula

Let's use the slope formula to find the slope of the line that goes through the points left parenthesis, 2, comma, 1, right parenthesis and left parenthesis, 4, comma, 7, right parenthesis.

Step 1: Identify the values of x, start subscript, 1, end subscript, x, start subscript, 2, end subscript, y, start subscript, 1, end subscript, and y, start subscript, 2, end subscript.

x, start subscript, 1, end subscript, equals, 2

y, start subscript, 1, end subscript, equals, 1

x, start subscript, 2, end subscript, equals, 4

y, start subscript, 2, end subscript, equals, 7, space, space, space, space, space, space, space, space
[Explain]

Step 2: Plug in these values to the slope formula to find the slope.

start text, S, l, o, p, e, end text, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end fraction, equals, start fraction, 7, minus, 1, divided by, 4, minus, 2, end fraction, equals, start fraction, 6, divided by, 2, end fraction, equals, 3

Step 3: Gut check. Make sure this slope makes sense by thinking about the points on the coordinate plane.

Yup! This slope seems to make sense since the slope is positive, and the line is increasing.

Using the slope formula walkthrough

Let's use the slope formula to find the slope of the line that goes through the points left parenthesis, 6, comma, minus, 3, right parenthesis and left parenthesis, 1, comma, 7, right parenthesis.

Step 1: Identify the values of x, start subscript, 1, end subscript, x, start subscript, 2, end subscript, y, start subscript, 1, end subscript, and y, start subscript, 2, end subscript.

x, start subscript, 1, end subscript, equals

y, start subscript, 1, end subscript, equals

x, start subscript, 2, end subscript, equals

y, start subscript, 2, end subscript, equals

[Show solution]

Step 2: Plug in these values to the slope formula to find the slope.

start text, S, l, o, p, e, end text, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end fraction, equals

[Show solution]

Step 3: Gut check. Make sure this slope makes sense by thinking about the points on the coordinate plane.

Does this slope make sense?

Let's practice!

1) Use the slope formula to find the slope of the line that goes through the points left parenthesis, 2, comma, 5, right parenthesis and left parenthesis, 6, comma, 8, right parenthesis.

[Show solution]

2) Use the slope formula to find the slope of the line that goes through the points left parenthesis, 2, comma, minus, 3, right parenthesis and left parenthesis, minus, 4, comma, 3, right parenthesis.

[Show solution]

3) Use the slope formula to find the slope of the line that goes through the points left parenthesis, minus, 5, comma, minus, 7, right parenthesis and left parenthesis, minus, 2, comma, minus, 1, right parenthesis.

[Show solution]

Something to think about

What happens in the slope formula when x, start subscript, 2, end subscript, equals, x, start subscript, 1, end subscript?

As a reminder, here is the slope formula:

start text, S, l, o, p, e, end text, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end fraction

Feel free to discuss in the comments below!

Want to join the conversation?

Sort by:

Jack Smith
Posted 6 years ago. Direct link to Jack Smith's post “Why is the slope formula ...”
Why is the slope formula y/x? Why not y-x or y+x?

Thanks to anyone who answers.
Jack.
Button navigates to signup pageComment on Jack Smith's post “Why is the slope formula ...”
(21 votes)
Answer
- Sanjana Gurram
  Posted 6 years ago. Direct link to Sanjana Gurram's post “Slope is something that i...”
  Slope is something that is also referred to as the rate of change. For example, if you had a savings account that you deposited no money into initially but you deposit 20$ weekly, your rate of change, or slope for this problem would be 20. This is because your x-value in this situation would be the number of weeks passed since you have created your bank account, and the y-value is how much money you have deposited into your account, fully. Since you are looking at the rate of change between the weeks, you divide the change in y per week, 20, by 1 for the number of weeks. I hope this somewhat answers your question.
  Comment on Sanjana Gurram's post “Slope is something that i...”
  (47 votes)
UAO
Posted 3 years ago. Direct link to UAO's post “*Something To Think About...”
Something To Think About
I think that when x_2 = x_1 then the slope will become undefined because x_2 - x_1 equals zero. Therefore when you divide y_2 - y_1 it won't be possible.

Example

(5,10) (5,15)

x_1 = 5
x_2 = 5
y_1 =10
y_2 =15

5 - 5= 0

15-10=5

5/0= Undefined
Button navigates to signup pageComment on UAO's post “*Something To Think About...”
(33 votes)
Answer
- Ian Pulizzotto
  Posted 3 years ago. Direct link to Ian Pulizzotto's post “Yes, you are correct. Th...”
  Yes, you are correct. The slope of any line through two different points with the same x-coordinate (that is, a vertical line) is always undefined, for the reason you stated.
  Button navigates to signup page
  (13 votes)
Adrian Ambrosetti
Posted 4 years ago. Direct link to Adrian Ambrosetti's post “I think that when X2 = X1...”
I think that when X2 = X1, the slope is undefined
Button navigates to signup pageButton navigates to signup page
(17 votes)
Answer
- Isha
  Posted 2 years ago. Direct link to Isha's post “Yes! That is correct.”
  Yes! That is correct.
  Comment on Isha's post “Yes! That is correct.”
  (6 votes)
christian31994
Posted 6 years ago. Direct link to christian31994's post “If I Get The Right Answer...”
If I Get The Right Answer Then Why Do I Have To Simplify?
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Isha
  Posted 2 years ago. Direct link to Isha's post “Simplifying just makes it...”
  Simplifying just makes it easier to read/understand. It makes it more "simple." Although both are equal, it is just easier to work with if it's simplified afterwards.
  Comment on Isha's post “Simplifying just makes it...”
  (7 votes)
Eileen Chao
Posted 3 years ago. Direct link to Eileen Chao's post “For the last one, if x_1 ...”
For the last one, if x_1 equals x_2 it is undefined, this is because, from the other videos, it was said that if the two points have the same x when drawing the line, it will be straight up and down, with no slope, but those are called undefined as there really is an undefined slope to it
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
- Ian Pulizzotto
  Posted 3 years ago. Direct link to Ian Pulizzotto's post “Yes you are correct that ...”
  Yes you are correct that the slope is undefined if x_1 = x_2. Good job!
  Comment on Ian Pulizzotto's post “Yes you are correct that ...”
  (2 votes)
Isaiah softhal
Posted 3 years ago. Direct link to Isaiah softhal's post “bro im honestly so gone”
bro im honestly so gone
Button navigates to signup pageComment on Isaiah softhal's post “bro im honestly so gone”
(5 votes)
Answer
- T̷h̷e̷C̷o̷d̷i̷n̷g̷L̷e̷g̷e̷n̷d̷ INACTIVE
  Posted 3 years ago. Direct link to T̷h̷e̷C̷o̷d̷i̷n̷g̷L̷e̷g̷e̷n̷d̷ INACTIVE's post “Slope is basically the ch...”
  Slope is basically the change in the y direction divided by the change in the x direction. If you don't know graphs, you might want to learn that first.
  Button navigates to signup page
  (2 votes)
Sean Chai
Posted a year ago. Direct link to Sean Chai's post “Something to Think About:...”
Something to Think About:
When x1 = x2, it means that x1-x2=0. So therefore the formula will simplify to y/0. And as x/0 is undefined, the slope should also be undefined.
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
sebastian.lomaglio
Posted 2 months ago. Direct link to sebastian.lomaglio's post “When x2=x1 then the slope...”
When x2=x1 then the slope will always be undefined because (y2-y1)/0 will always be undefined
Button navigates to signup pageButton navigates to signup page
(5 votes)
Answer
Samuel Bernhardt
Posted 7 years ago. Direct link to Samuel Bernhardt's post “In the very last part, wh...”
In the very last part, why is the formula for slope delta y/delta x instead of delta x/ delta y?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Tams Fletcher
  Posted 7 years ago. Direct link to Tams Fletcher's post “This is because the equat...”
  This is because the equation that describes a line is y=mx+c.
  If we have the y and x values (as in the coordinates), and c is constant for both points (which if it is two point on one line, we know it is) than we can solve for m with algebra.
  If we have two coordinates on a line (x1,y1 =1,2) and (x2, y2 =3,6) we can solve for m as follows.
  (x2,y2) 6=m3+c
  -
  (x1,y1) 2=m1+c
  
  1st step: c-c =0
  we are left with
  6=m3
  -
  2=m1
  The first equation minus the second =
  4=2m
  But we want the slope (m) on one side so we can solve for M.
  4/2=m
  2=m which is your slope
  
  What you have done here is take y2 from y1 on the left, x2 from x1 on the right, then divided by x to get m on its own. We can do this in one step instead to get the slope by the equation
  (y2-y1)/(x2-x1)=m
  That is why you divide by x rather than Y.
  Button navigates to signup page
  (7 votes)
Joy Young
Posted 6 years ago. Direct link to Joy Young's post “in the formula mx+b=y, I ...”
in the formula mx+b=y, I understand that "m" is the slope and "b" is the y-intercept, but what is x and y?
Button navigates to signup pageComment on Joy Young's post “in the formula mx+b=y, I ...”
(3 votes)
Answer
- Alan Wang
  Posted 6 years ago. Direct link to Alan Wang's post “when you graph the line, ...”
  when you graph the line, mx+b=y and fill in the slope and y-intercept, the x and y represent points that are on the line that you graphed. For example, if the equation was 5x+10=y, you could create pairs of (x,y) coordinates by plugging in numbers for x and y. In this case, if x was 5, y would be 35 or vice versa. Based on this, you could say that (5,35) is a point on the line, 5x+10=y.
  Comment on Alan Wang's post “when you graph the line, ...”
  (4 votes)