how to find b in y=mx+b

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The y-intercept of an equation is a point where the graph of the equation intersects the Y-axis.^[1] There are several ways to find the y-intercept of an equation, depending on the starting information you have.

1
Write down the slope and point. ^[2] 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 y-intercept of this line using the steps below.
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Learn the slope-intercept 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 y-intercept.

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Substitute the slope in this equation. Write the slope-intercept equation, but instead of m, use the slope of your line.
- Example 1 (cont.): y = mx + b
  m = slope = 2
  y = 2x + b
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Replace 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
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Solve for b. Remember, b is the y-intercept 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 y-intercept of this line is -2.
6
Write this as a coordinate point. The y-intercept is the point where the line intersects with the y-axis. Since the y-axis is located at x = 0, the x coordinate of the y-intercept is always 0.
- Example 1 (cont.): The y-intercept is at y = -2, so the coordinate point is (0, -2).
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1
Write down the coordinates of both points. ^[3] This method covers problems that only tell you two points on a straight line.^[4] Write each point coordinate down in (x,y) form.
- Example 2: A straight line passes through points (-1, 2) and (3, -4). Find the y-intercept of this line using the steps below.
2
Calculate 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" ( ${\frac {rise}{run}}$ ).^[5] Here's how to find these two quantities from two points:
- "Rise" is the change in vertical distance, or the difference between the y-values of the two points.
- "Run" is the change in horizontal distance, or the difference between x-values of the same two points.
- Example 2 (cont.): The y-values of the two points are 2 and -4, so the rise is (-4) - (2) = -6.
  The x-values of the two points (in the same order) are 1 and 3, so the run is 3 - 1 = 2.
3
Divide rise by run to find the slope. Now that you know these two values, plug them into " ${\frac {rise}{run}}$ " to find the slope of the line.
- Example 2 (cont.): $slope={\frac {rise}{run}}={\frac {-6}{2}}=$ -3.
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Review the slope-intercept form. You can describe a straight line with the formula y = mx + b, where m is the slope and b is the y-intercept.^[6] Now that we know the slope m and a point (x,y), we can use this equation to solve for b, the y-intercept.
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Fit the slope and point into the equation. Take the equation in slope-intercept 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] 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.
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Solve for b. Now the only variable left in the equation is b, the y-intercept. Rearrange the equation so b is on one side, and you have your answer.^[8] Remember, the y-intercept always has an x-coordinate of 0.
- Example 2 (cont.): 2 = -3(1) + b
  2 = -3 + b
  5 = b
  The y-intercept is at (0,5).
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1
Write down the equation of the line. If you already have the equation of the line, you can find the y-intercept with a little algebra.^[9]
- Example 3: What is the y-intercept 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).
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Substitute 0 for x. The y-axis is a vertical line along x = 0. This means any point on the y-axis has an x-coordinate of 0, including the y-intercept 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
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Solve for y. The answer is the y-intercept of the line.
- Example 3 (cont.): 4y = 16
  ${\frac {4y}{4}}={\frac {16}{4}}$
  y = 4.
  The y-intercept of the line is 4.
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Confirm by graphing (optional) . To check your answer, graph the equation as neatly as you can. The point where the line crosses the y-axis is the y-intercept.
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Find the y-intercept 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 y-axis at 0, 1, or 2 points. This means you may end up with 0, 1, or 2 answers.

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Question

If I have (3, -12) as my points and an x-intercept of one, what would the y-intercept be?

Solve 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 x-intercept, which is (1,0).
Question

If I am only given one specific point, how can I convert it into slope intercept form?

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Community Answer

You 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.
Question

I do not get the last part of method 2, can you explain?

Using the "slope-intercept" form of the line's equation (y = mx + b), you solve for b (which is the y-intercept you're looking for). Substitute the known slope for m, and substitute the known point's coordinates for x and y, respectively, in the slope-intercept 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 slope-intercept equation becomes 2 = -3(1) + b. So 2 = -3 + b, and b = 5. That's the y-intercept.
Question

What do I do if the only thing on the other side of the = is a variable?

If 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 y-intercept (b) of zero.
Question

How do I find the y intercept when it is not numbered and obviously a decimal?

Assuming by numbered you mean whole number, it doesn't matter, the method is the same in both cases.
Question

How do I find the y-intercept if 3x - y = 6?

The graph crosses the y-axis 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 y-intercept is -6.
Question

How do I find the y-intercept if the only given information is the slope and no graph or point(s) specified?

You would have to be given more information than just the slope in order to find the y-intercept.
Question

How do I find the slope for this and then the y-intercept (0,0) and (4,-3)?

The 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 y-intercept is the y-value when x=0. In this case, we are told that the line passes through the point (0,0), which means that the y-intercept is zero.
Question

What if the slope is undefined?

If 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 y-axis).
Question

Where is the y-axis on the Cartesian plane?

The y-axis is the vertical line that passes through the origin. It is the series of all points where x = 0.

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For more complicated equations, try to isolate the terms containing y onto one side of the equation.
Some countries use a c or another variable instead of b in the equation y = mx + b.^[10] This doesn't change the meaning; it's just a different tradition.
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] For example, the slope between (1, 12) and (3, 7) can be calculated in two different ways:

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Article SummaryX

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 slope-intercept 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 y-intercept if you only know 2 points along the line, keep reading the article!

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