How to Find the Equation of a Line

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Finding the equation for a line is a common problem in geometry and trigonometry. There are two common situations where you are asked to find the equation for a line: either you'll be provided with one point on the line and the slope of the line, or you'll be provided two points on the line. In either case, finding the equation for that line isn't difficult, provided you use the correct formula and work carefully.

Method 1
Method 1 of 2:

Calculating the Equation with One Point and the Slope

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    This is known as the point-slope formula. [1] The point-slope formula uses the slope and the coordinates of a point along the line to find the y-intercept. Use the slope in place of m in y-y 1 = m (x-x 1 ). [2]
    • For example, if you know the slope of the line is 2, then your formula will look like this: y-y 1 = 2(x-x 1 ) .
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    Use the coordinates you’re given as (x 1 , y 1 ). Put the numbers in the corresponding spot on your formula before you start solving the equation. [3]

    For example, if you know the coordinate is (4, 3), your formula will read: y-3 = 2(x-4) .

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    Follow the mathematical order of operations and the distributive property to remove the x-term from parenthesis. [4]
    • In our example, first you’d use the distributive property to get y-3=2x-8.
    • Then, add 3 to each side so y is by itself.
    • The final equation for a line in slope-intercept form with a slope of 2 that contains the point (4, 3) is y = 2x-5 .
    EXPERT TIP

    Joseph Meyer

    Math Teacher
    Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.
    Joseph Meyer
    Math Teacher

    The distributive property helps you avoid repetitive calculations. You can use the distributive property to solve equations where you must multiply a number by a sum or difference. It simplifies calculations, enables expression manipulation (like factoring), and forms the basis for solving many equations.

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Method 2
Method 2 of 2:

Finding the Equation Using Two Points

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    The ordered pairs of the coordinates you have are listed as (x, y). Use the first set of coordinates as (x 1 , y 1 ) and the second set as (x 2 , y 2 ). [5] Plug the numbers into the formula m = (y 2 -y 1 )/(x 2 -x 1 ) and solve for m . [6]

    For example, if your coordinates are (3, 8) and (7, 12), the formula would read: m = (12-8)/(7-3) = 4/4 = 1 . In this case, your slope, or m , equals 1.

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    The slope-intercept formula of a line is written as y = m x+b, where m is the slope and b is the y-intercept (the point on the y-axis where the line crosses it). Plug the number you found for your slope in place of m . [7]
    • In our example, the formula would read y = 1x+b or y = x+b when you replace the slope value.
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    Pick one of the ordered pairs to put into the slope-intercept formula. Put the x-value in place of x and the y-value in place of y. [8]
    • In this example, if you chose (3, 8) as your coordinates, then the formula would read 8 = 1(3)+b .
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    Once you plug the x- and y-values as well as your slope into the formula, find the value of b in the equation. Follow the order of operations first before moving the rest of the numbers to the other side. Leave b on one side of the equation to solve it. [9]
    • In our example, the formula currently reads 8 = 1(3)+b. Multiply 1 and 3 together to get 8 = 3+b. Since 3 is a positive number, subtract 3 from each side to isolate b. This leaves you with 5 = b, or b = 5.
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    Once you’re finished, plug in the slope for m and the y-intercept for b. After that, you’ve found the equation for the line. [10]
    • For example, the equation for the line with points on (3, 8) and (7, 12) is y = 1x+5 or simply y = x+5 .
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  • Question
    How do I find an intercept of a line?
    Donagan
    Top Answerer
    Using the equation, set x equal to zero and solve for y to find the y-intercept, or set y equal to zero and solve for x to find the x-intercept.
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  • Question
    What should I do with a negative y?
    Community Answer
    You multiply the whole equation by -1 to remove the negative sign. For example, if the question is: -y=-5x+1, you would then change the question to: -y*(-1) = (-5x+1)*(-1), and then get: y=5x-1.
    Thanks! We're glad this was helpful.
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  • Question
    What about given a slope and an intercept in y?
    Donagan
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    Express the equation in standard form, y = mx + b, where m is the slope, and b is the y-intercept.
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    Thank you for your feedback.
    If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
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      References

      1. ↑ Grace Imson, MA. Math Teacher. Expert Interview. 1 November 2019.
      2. ↑ http://www.coolmath.com/algebra/08-lines/11-finding-equation-line-point-slope-01
      3. ↑ Grace Imson, MA. Math Teacher. Expert Interview. 1 November 2019.
      4. ↑ Grace Imson, MA. Math Teacher. Expert Interview. 1 November 2019.
      5. ↑ https://www.cuemath.com/geometry/equation-of-a-line/
      6. ↑ Grace Imson, MA. Math Teacher. Expert Interview. 1 November 2019.
      7. ↑ https://www.cuemath.com/geometry/equation-of-a-line/
      8. ↑ https://content.byui.edu/file/b8b83119-9acc-4a7b-bc84-efacf9043998/1/Math-2-11-2.html
      9. ↑ https://content.byui.edu/file/b8b83119-9acc-4a7b-bc84-efacf9043998/1/Math-2-11-2.html
      1. ↑ https://openstax.org/books/intermediate-algebra-2e/pages/3-3-find-the-equation-of-a-line

      About This Article

      Article Summary X

      To find the equation of a line using 2 points, start by finding the slope of the line by plugging the 2 sets of coordinates into the formula for slope. Then, plug the slope into the slope-intercept formula, or y = mx + b, where "m" is the slope and "x" and "y" are one set of coordinates on the line. Next, solve the formula to find the value of "b," which is the y-intercept. Finally, plug the slope and y-intercept into the slope-intercept formula to finish the equation of the line. To learn how to find the equation of a line using 1 point, scroll down!

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