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 = \frac{∆y}{∆x} \\ \,\\ = \frac{y_2 - y_1}{x_2 - x_1}

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

m = \frac{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:

    Ax + By + C = 0
    By = -Ax - C \\ \,\\ y = -\frac{A}{B}x - \frac{C}{B}

    The equation

    y = -\frac{A}{B}x - \frac{C}{B}

    has the form

    y = mx +b


    m = - \frac{A}{B}


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

-\frac{A}{B} = - \frac{2}{3}

Example 2​: What is the slope of the line

x = \frac{3}{7}y -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.

    \frac{3}{7}y = x + 22
    3y = 7x + 154
    y = \frac{7}{3}x + 51.33

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

    m = \frac{7}{3}

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