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Writing slope-intercept equations

Learn how to find the slope-intercept equation of a line from two points on that line.
If you haven't read it yet, you might want to start with our introduction to slope-intercept form.

Writing equations from y-intercept and another point

A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points zero, three and two, seven, which are plotted and labeled.
Let's write the equation of the line that passes through the points left parenthesis, 0, comma, 3, right parenthesis and left parenthesis, 2, comma, 7, right parenthesis in slope-intercept form.
Recall that in the general slope-intercept equation y, equals, start color #ed5fa6, m, end color #ed5fa6, x, plus, start color #0d923f, b, end color #0d923f, the slope is given by start color #ed5fa6, m, end color #ed5fa6 and the y-intercept is given by start color #0d923f, b, end color #0d923f.

Finding start color #0d923f, b, end color #0d923f

The y-intercept of the line is left parenthesis, 0, comma, start color #0d923f, 3, end color #0d923f, right parenthesis, so we know that start color #0d923f, b, end color #0d923f, equals, start color #0d923f, 3, end color #0d923f.

Finding start color #ed5fa6, m, end color #ed5fa6

Recall that the slope of a line is the ratio of the change in y over the change in x between any two points on the line:
start text, S, l, o, p, e, end text, equals, start fraction, start text, C, h, a, n, g, e, space, i, n, space, end text, y, divided by, start text, C, h, a, n, g, e, space, i, n, space, end text, x, end fraction
Therefore, this is the slope between the points left parenthesis, 0, comma, 3, right parenthesis and left parenthesis, 2, comma, 7, right parenthesis:
m=Change in yChange in x=7320=42=2\begin{aligned}\maroonC{m}&=\dfrac{\text{Change in }y}{\text{Change in }x} \\\\ &=\dfrac{7-3}{2-0} \\\\ &=\dfrac{4}{2} \\\\ &=\maroonC{2}\end{aligned}
In conclusion, the equation of the line is y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, start color #0d923f, plus, 3, end color #0d923f.

Check your understanding

Problem 1
A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points zero, five and four, nine, which are plotted and labeled.
Write the equation of the line.

Problem 2
A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points zero, eight and three, two which are plotted and labeled.
Write the equation of the line.

Writing equations from any two points

A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points two, five and four, nine, which are plotted and labeled.
Let's write the equation of the line that passes through left parenthesis, 2, comma, 5, right parenthesis and left parenthesis, 4, comma, 9, right parenthesis in slope-intercept form.
Note that we are not given the y-intercept of the line. This makes things a little bit more difficult, but we are not afraid of a challenge!

Finding start color #ed5fa6, m, end color #ed5fa6

m=Change in yChange in x=9542=42=2\begin{aligned} \maroonC{m}&=\dfrac{\text{Change in }y}{\text{Change in }x} \\\\ &=\dfrac{9-5}{4-2} \\\\ &=\dfrac{4}{2} \\\\ &=\maroonC{2} \end{aligned}

Finding start color #0d923f, b, end color #0d923f

We know that the line is of the form y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, plus, start color #0d923f, b, end color #0d923f, but we still need to find start color #0d923f, b, end color #0d923f. To do that, we substitute the point left parenthesis, 2, comma, 5, right parenthesis into the equation.
Because any point on a line must satisfy that line’s equation, we get an equation that we can solve to find start color #0d923f, b, end color #0d923f.
y=2x+b5=22+bx=2 and y=55=4+b1=b\begin{aligned}y&=\maroonC{2}\cdot x+\greenE{b}\\\\ 5&=\maroonC{2}\cdot 2+\greenE{b}&\gray{x=2\text{ and }y=5}\\\\ 5&=4+\greenE{b}\\\\ \greenE{1}&=\greenE{b} \end{aligned}
In conclusion, the equation of the line is y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, start color #0d923f, plus, 1, end color #0d923f.

Check your understanding

Problem 3
A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points one, four and three, ten, which are plotted and labeled.
Write the equation of the line.

Problem 4
A first quadrant coordinate plane. The x- and y-axes each scale by one. A graph of a line goes through the points two, nine and four, one, which are plotted and labeled.
Write the equation of the line.

Challenge problem
A line passes through the points left parenthesis, 5, comma, 35, right parenthesis and left parenthesis, 9, comma, 55, right parenthesis.
Write the equation of the line.

Want to join the conversation?

  • blobby green style avatar for user ahernandez26
    i think i'll just sell corn on the street
    (18 votes)
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  • blobby green style avatar for user 4029212
    Bruh this is hard to do. I still dont undertand
    (13 votes)
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  • starky seed style avatar for user NicholasT
    no way that this makes a single drop of sense
    (7 votes)
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  • blobby green style avatar for user Abigail A layla:)
    bro why does hurt my brain
    like i did fine on my Quiz
    (5 votes)
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  • piceratops seedling style avatar for user ReverseSwitch09
    Math teacher who?! Khan Acadamy is so much better
    (5 votes)
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  • leaf green style avatar for user Anna
    On number 4, why would b=17?
    (2 votes)
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    • leaf green style avatar for user Diya
      This is the process they show you:
      y =−4x+b
      9 =−4⋅2+b
      9 =−8+b
      17=b
      ----------
      In THIS ("9 =−8+b") step they are subtracting -8 from 9 which looks like this: 9-(-8). We know that two negative operations make a positive sum, which is why the answer is 17 : 9-(-8) = 17.

      (2 votes)
  • blobby green style avatar for user laurie
    So I get how to get the y-intercept but I'm having hard time with the rest. Any ideas to help me out please and thank you.
    (2 votes)
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    • stelly yellow style avatar for user ⭐BEST20042007⭐
      Hey there Laurie! So let's just say we are given two random points which are (31,5) and (33,27). We have to figure out an slope intercept for equation for it. Well, we already know that the format of the equation should look like this: y = mx + b.

      So first, let's start out with finding the slope. When trying to find the slope, we first have to use the common formula y change over x change. Or in other words, rise over run. Now, lets take our two points and minus the y's. Final answer: 27 - 5 = 22. Good, now just remember you can subtract the points any way you want, but most people just prefer to subtract y2 - y1. That's what we did. Moving on, let's apply the same method we just used to subtract the x's. Final answer: 33 - 31 = 2.

      Great! So now we are left with out a y that is 22 and a x that is 2. This is the step in which we apply the strategy y change over x change. All we have to do now is just divide the y by the x! Final answer: 22/2 = 11. 11 is now our slope!

      Status Check: Your current equation should look like this so far: *y = 11x

      Following, when we add our slope (11) to our equation, please be sure to also include the x being multiplied to it as well. So here, you said you know how to get the y-int, but I will just go over it very quick! With our two given points on a graph, just pick any one. It doesn't really matter. I will just go with the point (31,5). Now, with the x and y in this point, we must substitute it for the x and y in the equation. With the substituted numbers in the equation, it should look like this so far: 5 = 11(31) + b Last but not least, don't forget to solve! Multiply 11 by 31 and get 341. And just subtract that from 5 and you will get -336! That will be your y intercept!

      To sum up, this is basically how to write a slope intercept equation with 2 given points! Final answer: y = 11x - 336 If you have any questions regarding this, please feel free to ask!! I hope this helps.
      (3 votes)
  • male robot hal style avatar for user John Malig
    On the challenge problem when I had to solve for b, instead of using y=mx+b,

    I used b=(0,?) (5,35) = (0,10)
    ---------------(-5,-5(5))

    I subtracted 5 from x so as to make it 0, and multiplied that subtraction to the slope to solve for y. Is that method valid?
    (2 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      Yes, it's a completely valid method.

      In order to get from (5, 35) to (0, 𝑏) we would have to move −5 along the 𝑥-axis, which would also move us 5 ∙ (−5) = −25 along the 𝑦-axis.
      Thereby, (0, 𝑏) = (5, 35) + (−5, −25) = (0, 10) ⇒ 𝑏 = 10

      Good job!
      (3 votes)
  • hopper cool style avatar for user Yo Algebra
    This is so hard How do you do number four
    (3 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      First find the slope: rise/run = (9-1)/(2-4) = 8/(-2) = -4.

      So the equation so far is y = -4x + b.

      Now substitute either point and solve for b. Let’s choose the point (4,1). So we have
      1 = -4*4 + b
      1 = -16 + b
      b = 17.

      The equation is y = -4x + 17.

      It’s a good idea to check the equation by verifying that both given points make the equation true.
      The point (4,1) makes the equation true because 1 = -4*4 + 17.
      The point (2,9) makes the equation true because 9 = -4*2 + 17.

      Therefore, since two points determine a line, the equation y = -4x + 17 is correct.

      Have a blessed, wonderful Valentine’s Day!
      (1 vote)
  • spunky sam orange style avatar for user daP0l15hc0unt
    Is there a way to write the equation of a line using the x-intercept and the slope?
    (2 votes)
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