How To Find Slope From An Equation

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 (​x​₁, ​y​₁) and (​x​₂, ​y​₂) 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:

1. Write the Equation in Standard Form

\(Ax + By + C = 0\)

2. Rearrange to Get y by Itself

\(By = -Ax – C \
\,\
y = -\frac{A}{B}x – \frac{C}{B}\)

3. Read Slope from the Equation

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.

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

\(\frac{3}{7}y = x + 22\)

2. Multiply Both Sides by 7

\(3y = 7x + 154\)

3. Divide Both Sides by 3

\(y = \frac{7}{3}x + 51.33\)

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

\(m = \frac{7}{3}\)