How to Find Slope From an Equation

••• jacoblund/iStock/GettyImages

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

    where

    m = - \frac{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

-\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}

Related Articles

How to Find B in Y=Mx + B
Standard Form of a Linear Equation
How to Solve Slope-Intercept Form
How to Convert Meters per Second to Miles per Hour
Standard Form of a Linear Equation
How to Convert an Equation Into Vertex Form
How to Find B in Y=Mx + B
How to Interpret Linear Equations
How to Find Perpendicular Slope
How to Find a Parallel Line
How to Calculate the Slope of a Tangent
How to Find the Slope of Linear Equations
How to Determine the Y-Intercept of a Trend Line
How to Find the Slope of a Nonlinear Line
How to Convert Negative Celsius to Fahrenheit
How to Tell If Lines Are Parallel, Perpendicular or...
How to Convert Point Slope Form to Slope Intercept...
How to Convert Angle Degrees to Slope
How to Find Equations of Tangent Lines

Dont Go!

We Have More Great Sciencing Articles!