This article was coauthored by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University.
There are 8 references cited in this article, which can be found at the bottom of the page.
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The yintercept of an equation is a point where the graph of the equation intersects the Yaxis.^{[1] X Research source } There are several ways to find the yintercept of an equation, depending on the starting information you have.
Steps
Method 1
Method 1 of 3:Finding the YIntercept from the Slope and Point

1Write down the slope and point.^{[2] X Research source } The slope or "rise over run" is a single number that tells you how steep the line is. This type of problem also gives you the (x,y) coordinate of one point along the graph. Skip to the other methods below if you don't have both these pieces of information.
 Example 1: A straight line with slope 2 contains the point (3,4). Find the yintercept of this line using the steps below.

2Learn the slopeintercept form of an equation. Any straight line can be written as an equation in the form y = mx + b. When the equation is in this form, the variable m is the slope, and b is the yintercept.Advertisement

3Substitute the slope in this equation. Write the slopeintercept equation, but instead of m, use the slope of your line.

Example 1 (cont.): y = mx + b
m = slope = 2
y = 2x + b

Example 1 (cont.): y = mx + b

4Replace x and y with the coordinates of the point. Any time you have the coordinates of a single point on your line, you can substitute those x and y coordinates for the x and y in your line equation. Do this for the equation you've been working on.

Example 1 (cont.): The point (3,4) is on this line. At this point, x = 3 and y = 4.
Substitute these values into y = 2x +b:
4 = 2(3) + b

Example 1 (cont.): The point (3,4) is on this line. At this point, x = 3 and y = 4.

5Solve for b. Remember, b is the yintercept of the line. Now that b is the only variable in the equation, rearrange to solve for this variable and find the answer.

Example 1 (cont.): 4 = 2(3) + b
4 = 6 + b
4  6 = b
2 = b
The yintercept of this line is 2.

Example 1 (cont.): 4 = 2(3) + b

6Write this as a coordinate point. The yintercept is the point where the line intersects with the yaxis. Since the yaxis is located at x = 0, the x coordinate of the yintercept is always 0.
 Example 1 (cont.): The yintercept is at y = 2, so the coordinate point is (0, 2).
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Method 2
Method 2 of 3:Using Two Points

1Write down the coordinates of both points.^{[3] X Research source } This method covers problems that only tell you two points on a straight line.^{[4] X Research source } Write each point coordinate down in (x,y) form.
 Example 2: A straight line passes through points (1, 2) and (3, 4). Find the yintercept of this line using the steps below.

2Calculate the rise and run. Slope is a measure of how much vertical distance the line moves for each unit of horizontal distance. You may have heard this described as "rise over run" ().^{[5] X Expert Source Grace Imson, MAMath Teacher Expert Interview. 1 November 2019. } Here's how to find these two quantities from two points:
 "Rise" is the change in vertical distance, or the difference between the yvalues of the two points.
 "Run" is the change in horizontal distance, or the difference between xvalues of the same two points.

Example 2 (cont.): The yvalues of the two points are 2 and 4, so the rise is (4)  (2) = 6.
The xvalues of the two points (in the same order) are 1 and 3, so the run is 3  1 = 2.

3Divide rise by run to find the slope. Now that you know these two values, plug them into "" to find the slope of the line.
 Example 2 (cont.): 3.

4Review the slopeintercept form. You can describe a straight line with the formula y = mx + b, where m is the slope and b is the yintercept.^{[6] X Expert Source Grace Imson, MAMath Teacher Expert Interview. 1 November 2019. } Now that we know the slope m and a point (x,y), we can use this equation to solve for b, the yintercept.

5Fit the slope and point into the equation. Take the equation in slopeintercept form and replace m with the slope you calculated. Replace the x and y terms with the coordinates of a single point on the line.^{[7] X Expert Source Grace Imson, MAMath Teacher Expert Interview. 1 November 2019. } It doesn't matter which point you use.

Example 2 (cont.): y = mx + b
Slope = m = 3, so y = 3x + b
The line includes a point with (x,y) coordinates (1,2), so 2 = 3(1) + b.

Example 2 (cont.): y = mx + b

6Solve for b. Now the only variable left in the equation is b, the yintercept. Rearrange the equation so b is on one side, and you have your answer.^{[8] X Expert Source Grace Imson, MAMath Teacher Expert Interview. 1 November 2019. } Remember, the yintercept always has an xcoordinate of 0.

Example 2 (cont.): 2 = 3(1) + b
2 = 3 + b
5 = b
The yintercept is at (0,5).
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Example 2 (cont.): 2 = 3(1) + b
Method 3
Method 3 of 3:Using an Equation

1Write down the equation of the line. If you already have the equation of the line, you can find the yintercept with a little algebra.^{[9] X Research source }
 Example 3: What is the yintercept of the line x + 4y = 16?
 Note: Example 3 is a straight line. See the end of this section for an example of a quadratic equation (with a variable raised to the power of 2).

2Substitute 0 for x. The yaxis is a vertical line along x = 0. This means any point on the yaxis has an xcoordinate of 0, including the yintercept of the line. Plug in 0 for x in the line equation.

Example 3 (cont.): x + 4y = 16
x = 0
0 + 4y = 16
4y = 16

Example 3 (cont.): x + 4y = 16

3Solve for y. The answer is the yintercept of the line.

Example 3 (cont.): 4y = 16
y = 4.
The yintercept of the line is 4.

Example 3 (cont.): 4y = 16

4Confirm by graphing (optional). To check your answer, graph the equation as neatly as you can. The point where the line crosses the yaxis is the yintercept.

5Find the yintercept for a quadratic equation. A quadratic equation includes a variable (x or y) raised to the power of 2. You can solve for y with the same substitution, but since the quadratic describes a curve, it could intercept the yaxis at 0, 1, or 2 points. This means you may end up with 0, 1, or 2 answers.

Example 4: To find the yintercept of , substitute x = 0 and solve the quadratic equation.
In this case, we can solve by taking the square root of both sides. Remember, when taking a square root, you must account for two answers: a negative and a positive.
y = 1 or y = 1. These are both yintercepts of this curve.
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Example 4: To find the yintercept of , substitute x = 0 and solve the quadratic equation.
Community Q&A

QuestionIf I have (3, 12) as my points and an xintercept of one, what would the yintercept be?DonaganTop AnswererSolve by using Method 2 above. You need two points to use that method. One point is given as (3,12). The second point is the given xintercept, which is (1,0).

QuestionIf I am only given one specific point, how can I convert it into slope intercept form?TechnistCommunity AnswerYou can't have only one point. Imagine you're standing in the middle of your neighborhood street and have no where to go. No direction, no distance, etc. You don't need to go to your friend's house or to school. That is an example of having only one point without any other point to go to so there's no line. Thus, no equation. And more importantly, no math to work out.

QuestionI do not get the last part of method 2, can you explain?DonaganTop AnswererUsing the "slopeintercept" form of the line's equation (y = mx + b), you solve for b (which is the yintercept you're looking for). Substitute the known slope for m, and substitute the known point's coordinates for x and y, respectively, in the slopeintercept equation. That will let you find b. In the example in Method 2, the calculated slope is 3, so m becomes 3. The known point is given as (1,2), so x in the equation becomes 1, and y becomes 2. Thus, the slopeintercept equation becomes 2 = 3(1) + b. So 2 = 3 + b, and b = 5. That's the yintercept.

QuestionWhat do I do if the only thing on the other side of the = is a variable?DonaganTop AnswererIf you mean that the only thing on the "x" side of the equation is x (so that y = x), that indicates a slope (m) of 1 and a yintercept (b) of zero.

QuestionHow do I find the y intercept when it is not numbered and obviously a decimal?Community AnswerAssuming by numbered you mean whole number, it doesn't matter, the method is the same in both cases.

QuestionHow do I find the yintercept if 3x  y = 6?Community AnswerThe graph crosses the yaxis when x=0, so substitute 0 for x in this equation, and solve for y: y is 6. Since y is 6 when x is 0, the yintercept is 6.

QuestionHow do I find the yintercept if the only given information is the slope and no graph or point(s) specified?DonaganTop AnswererYou would have to be given more information than just the slope in order to find the yintercept.

QuestionHow do I find the slope for this and then the yintercept (0,0) and (4,3)?DonaganTop AnswererThe slope has a numerator consisting of the difference between any two x values on the line and a denominator consisting of the difference between the two corresponding y values. In this case the slope is (4  0) / (3  0), or 4 / 3, which equals 4/3. The yintercept is the yvalue when x=0. In this case, we are told that the line passes through the point (0,0), which means that the yintercept is zero.

QuestionWhat if the slope is undefined?DonaganTop AnswererIf a slope is said to be "undefined," that means that the slope is infinite, which in turn means that the line is vertical (or parallel to the yaxis).

QuestionWhere is the yaxis on the Cartesian plane?DonaganTop AnswererThe yaxis is the vertical line that passes through the origin. It is the series of all points where x = 0.
Video
Tips
 For more complicated equations, try to isolate the terms containing y onto one side of the equation.Thanks!
 Some countries use a c or another variable instead of b in the equation y = mx + b.^{[10] X Research source } This doesn't change the meaning; it's just a different tradition.Thanks!
 When calculating slope between two points, you can subtract the x and y coordinates from each other in either order, as long as you put the points in the same order for both rise and run.^{[11] X Research source } For example, the slope between (1, 12) and (3, 7) can be calculated in two different ways:
 Second point  first point:
 First point  second point:
Thanks!
References
 ↑ http://www.math.com/school/subject2/lessons/S2U4L2GL.html
 ↑ https://www.khanacademy.org/math/algebrahome/alglineareqfunc/algwritingslopeinterceptequations/v/findingyinterceptgivenslopeandpoint
 ↑ https://www.khanacademy.org/math/algebra/twovarlinearequations/writingslopeinterceptequations/v/equationofaline3
 ↑ https://www.mathplanet.com/education/algebra1/formulatinglinearequations/writinglinearequationsusingtheslopeinterceptform
 ↑ Grace Imson, MA. Math Teacher. Expert Interview. 1 November 2019.
 ↑ Grace Imson, MA. Math Teacher. Expert Interview. 1 November 2019.
 ↑ Grace Imson, MA. Math Teacher. Expert Interview. 1 November 2019.
 ↑ Grace Imson, MA. Math Teacher. Expert Interview. 1 November 2019.
 ↑ http://www.webmath.com/equline3.html
About This Article
To find the y intercept using the equation of the line, plug in 0 for the x variable and solve for y. If the equation is written in the slopeintercept form, plug in the slope and the x and y coordinates for a point on the line to solve for y. If you don't know the slope, calculate it by dividing the rise of the line by the run. If you want to find the yintercept if you only know 2 points along the line, keep reading the article!