How to Find Slope From an Equation

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A linear equation is one that relates the first power of two variables, x and y, and its graph is always a straight line. The standard form of such an equation is

Ax + By + C = 0

where A, B and C are constants.

Every straight line has slope, usually designated by the letter m. Slope is defined as the change in y divided by the change in x between any two points (x1, y1) and (x2, y2) on the line.

m = ∆y/∆x = (y2 - y1) ÷ (x2 - x1)

If the line passes through point (a, b) and any other random point (x, y), slope can be expressed as:

m = (y - b) ÷ (x - a)

This can be simplified to produce the slope-point form of the line:

y - b = m(x - a)

The y-intercept of the line is the value of y when x = 0. The point (a, b) becomes (0, b). Substituting this into the slope-point form of the equation, you get the slope-intercept form:

y = mx + b

You now have all you need to find the slope of a line with a given equation.

General Approach: Convert from Standard to Slope-Intercept Form

If you have an equation in standard form, it takes just a few simple steps to convert it to slope intercept form. Once you have that, you can read slope directly from the equation:

  1. Write the Equation in Standard Form

  2. Ax + By + C = 0

  3. Rearrange to Get y by Itself

  4. By = -Ax - C

    y = -(A/B)x - (C/B)

  5. Read Slope from the Equation

  6. The equation y = -A/B x - C/B has the form y = mx +b, where

    m = -(A/B)

Examples

Example 1: What is the slope of the line 2x + 3y + 10 = 0?

In this example, A = 2 and B = 3, so the slope is -(A/B) = -2/3.

Example 2: What is the slope of the line x = 3/7y -22?

You can convert this equation to standard form, but if you're looking for a more direct method to find slope, you can also convert directly to slope intercept form. All you have to do is isolate y on one side of the equal sign.

  1. Add 22 to Both Sides and Put the y Term on the Right

  2. 3/7y = x + 22

  3. Multiply Both Sides by 7

  4. 3y = 7x + 154

  5. Divide Both Sides by 3

  6. y = (7/3)x + 51.33

    This equation has the form y = mx + b, and

    m = 7/3

References

About the Author

Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010, offering information in scientific, cultural and practical topics. His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts.

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