Slope Calculator

Use the slope calculator to find the slope of a line that passes through 2 points or solve a coordinate given m. Plus, the calculator will also solve the slope-intercept form of a line.

Find the Slope of a Line

Point #1 Coordinates:
Point #2 Coordinates:

Find Points on a Line at a Distance From Another Point

Point Coordinates:
or
degrees

Find a Point on a Line Given it's x or y Value

Point #1 Coordinates:
Point #2 x or y Coordinate:
or
Enter an x or y coordinate for the second point to find the other value
or
degrees

Slope:

m =
2
The slope of the line connecting (1, 3) and (2, 5) is 2

Line Graph:


Slope Intercept Form:

y = 2x + 1

Point Slope Form:

y − 3 = 2(x - 1)

Standard Form:

2x - 1y = -1

Angle, Distance, & Intercepts:

Angle (θ):
63.435°
Distance:
2.236
Δx:
1
Δy:
2
x-Intercept:
-0.5
y-Intercept:
1

Steps to Find Slope

Start with the slope formula

m=(y2 - y1) / (x2 - x1)

Step One: Substitute point values in the formula

m=(5 - 3) / (2 - 1)

Step Two: Simplify the fraction

m=(5 - 3) / (2 - 1)=2 / 1

Step Three: Solve for slope (m)

m=2
Learn how we calculated this below


On this page:


How to Find the Slope of a Line

Slope is a measure of the rate of incline or steepness of a line on a graph. You can find the slope of a line by comparing any 2 points on the line. A point is an x and y value of a cartesian coordinate on a grid.

Slope, expressed m, is equal to the rise between two coordinates on a line over the run, or rather, it’s the ratio of rise to run.

Rise is the vertical change of the line, or the change in altitude or elevation, and the run is the horizontal change, or the change in distance.[1]

Slope Formula

You can find the slope m of a line using the following formula:[2]

m = y2 – y1 / x2 – x1

Thus, the slope m is equal to y2 minus y1, divided by x2 minus x1.

Graphic showing the slope formula where m is equal to the rise over run, or y2 minus y1 divided by x2 minus x1

Steps to Find Slope Using the Formula

To find the slope of a line using this formula, you’ll need to find 2 points along a line and find their x and y values. The value of x is the horizontal distance of the point from the vertical y-axis, and y is the vertical distance of the point from the horizontal x-axis.

Graph showing a line and the slope formula to solve for m

Then, follow three easy steps to calculate the slope.

Step One: Substitute Points in the Slope Formula

Start by replacing the y1, y2, x1, and x2 values in the formula with the coordinates for the two points on the line.

Step Two: Find the Difference in x and y Coordinates

Next, solve the numerator and denominator of the slope formula using these values. The result will be the absolute value of the distance between the x values or y values of two points.

The numerator represents the change in the value of y, which is known as the delta of y, or just Δy. You can find the delta of y using the formula:

Δy = y2 – y1

The denominator represents the change in the value of x, which is known as the delta of x, or Δx. You can find the delta of x using the formula:

Δx = x2 – x1

Step Three: Divide the Delta of y by the Delta of x

The final step is to divide the delta of y by the delta of x to find the slope. You can do this by dividing the numerator in the slope formula by the denominator, or more simply by using a fraction to decimal calculator.

In some cases, you may want to represent the slope as a fraction or ratio. In this case, you can simplify the fraction to reduce it to its simplest form rather than converting it to a decimal.

For example, let’s find the slope of a line that passes through the points (3,2) and (11,8).

Start with the slope formula:

m = (y2 – y1) / (x2 – x1)

Replace the x and y values with the coordinate’s x and y values.

m = 8 – 2 / 11 – 3

Solve the numerator and denominator to find the delta of x and y.

m = 8 – 2 / 11 – 3 = 6 / 8

Then, reduce the fraction to its simplest form.

m = 6 / 8 = 3 / 4

So, the slope m is equal to 3 / 4

The slope formula is useful for points along a linear line, but when working with non-linear functions, you might need to use an average rate of change calculator.

How to Interpret Slope

While the slope of a line doesn’t tell us where the line is located on the graph, it does tell us the angle of the line.

Graphic showing the slope of a line when m is positive, a line when m is negative, a line when m is equal to zero, and a line when m is undefined

If the slope is positive, then the line is slanted up and rises from left to right on the graph.

If the slope is negative, then the line is slanted down and drops from left to right on the graph.

If the slope is equal to zero, the line is a horizontal line and does not slant up or down. This is a line with no slope.

In the special case of a vertical line, the slope is undefined. Essentially the slope would be infinite in this case, and because this would result in the denominator of the slope formula being zero since there is no change in x, it is undefined.

How to Find Equations of a Line Using the Slope

Now that you found the slope, you can express a linear line using one of several equations. The most common equations for a line are slope-intercept form, point-slope form, and standard form.

How to Find Slope-Intercept Form

Slope-intercept form is one of the most commonly used equations to represent a linear line. You can find the equation of a line using slope-intercept form given the slope m and one point on the line.

Slope-Intercept Form Equation

The slope-intercept form equation is:[3]

y = mx + b

The slope-intercept form equation states that the y value of a coordinate on the line is equal to the x value of the coordinate times the slope m, plus the y-intercept b.

The slope-intercept form equation for a linear line showing that y equals mx plus b

For example, let’s solve for b, given a slope of 1/2 and a point (5,4). We’ll express this line in slope-intercept form.

Substitute for m:

y = 1 / 2 × x + b

Substitute the point values for x and y:

4 = 1 / 2 × 5 + b

Solve for b:

4 = 2.5 + b
4 – 2.5 = b
1.5 = b

Substitute b in the equation to find the slope-intercept form:

y = 1 / 2x + 1.5

How to Find Point-Slope Form

Point-slope form is another commonly used equation to represent a line. You can find the equation of a line using point-slope form given the slope m and one point on the line.

Point-Slope Form Equation

The point-slope form equation is:[4]

y – y1 = m(x – x1)

The point-slope form equation states that y minus the y1 coordinate is equal to the slope m times x minus the x1 coordinate.

The point-slope form equation for a line showing that y minus y1 is equal to m times x minus x1

For example, let’s express the same line from the previous example using point-slope form. Recall the line has a slope of 1/2 and a point (5,4).

Substitute for m:

y – y1 = 1 / 2(x – x1)

Substitute the point values for x and y:

y – 4 = 1 / 2(x – 5)

So, the point-slope form for this line is y – 4 = 1 / 2(x – 5).

How to Find Standard Form

Standard form is a standardized format for linear equations. To use standard form, the slope and coordinates must be whole numbers; decimal values are not supported in standard form.

Standard Form Equation

The standard form equation is:

Ax + By = C

Given either of the line equations above, you can convert to standard form by simply rearranging the equations to isolate C on one half of the equation. If there are any fractions, then multiply both sides of the equation by the denominator of the fractions to remove them.

The standard form for linear equations showing that Ax plus By is equal to C

This formula represents the standard form for linear equations, but numbers can also be expressed in standard form as well.

How to Find the x and y Intercepts of a Line

Given the slope of a line and one point on it, you can find the x and y intercepts using one of the line equations above. Start by finding the slope-intercept form equation for the line.

Then, given slope-intercept form, you can find the y-intercept by setting x in the equation to 0 and solving.

And to find the x-intercept, set y in the equation to 0 and solve it using algebra.

For example, let’s find the x and y intercepts of the line y = 1 / 2x + 8.

Start by setting x to 0.

y = (1 / 2 × 0) + 8

Then solve for y.

y = 8

When x is equal to zero y is equal to 8, so 8 is the y-intercept.

Next, set y to 0.

0 = 1 / 2x + 8

Then solve for x.

0 = 1 / 2x + 8
0 – 8 = 1 / 2x + 8 – 8
-8 = 1 / 2x
-8 / 1 = 1 / 2x
-8 × 2 / 1 = 1 × 2 / 2x
-16 / 1 = 2 / 2x
-16 = x

When y is equal to zero, x is equal to -16, so the x-intercept is -16.

How to Find a Point Given 1 Point, Slope, and Distance

Points on a line can be solved given the slope of the line and the distance from another point. The formulas to find x and y of the point to the right of the point are:

x2 = x1 + d / (1 + m2)

y2 = y1 + m × d / (1 + m2)

The formulas to find x and y of the point to the left of the point are:

x2 = x1 + -d / (1 + m2)

y2 = y1 + m × -d / (1 + m2)

How to Convert Slope to Angle

The angle of a line in degrees can be found from the inverse tangent of the slope m.

θ = tan-1(m)

For example, if m = 5, then the angle in degrees is tan-1(5).

Our rise and run to degrees converter can help calculate the value in degrees given the rise and run of a line.

How to Convert Angle to Slope

It’s also possible to convert an angle in degrees to slop, which is equal to the tangent of the angle.

m = tan(θ)

For example, if angle = 72, then m is equal to tan(72).

How to Find the Distance Between 2 Points

The formula to find the distance d between two points on a line is:

d = √((x2 – x1)2 + (y2 – y1)2)

One real-world example of using slope is to calculate the grade for landscaping and road-work projects.

References

  1. LibreTexts Mathematics, Your Guide to Intermediate Algebra - 3.3: Slope Intercept Form, https://math.libretexts.org/Courses/Kansas_State_University/Your_Guide_to_Intermediate_Algebra/03%3A_Linear_Equations_and_Graphs/3.03%3A_Slope_Intercept_Form
  2. Marecek, L., Mathis, A., Intermediate Algebra 2e - 3.2 Slope of a Line, OpenStax, https://openstax.org/books/intermediate-algebra-2e/pages/3-2-slope-of-a-line
  3. CK-12 Foundation, Linear Equations in Slope-Intercept Form, https://www.ck12.org/book/ck-12-algebra-basic/section/5.1/
  4. Clapham, C., Nicholson, J., Oxford Concise Dictionary of Mathematics, 475. https://web.archive.org/web/20131029203826/http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf