# 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

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

### Cite This Article

#### MLA

Deziel, Chris. "How To Find Slope From An Equation" *sciencing.com*, https://www.sciencing.com/how-to-find-slope-from-an-equation-13712210/. 2 November 2020.

#### APA

Deziel, Chris. (2020, November 2). How To Find Slope From An Equation. *sciencing.com*. Retrieved from https://www.sciencing.com/how-to-find-slope-from-an-equation-13712210/

#### Chicago

Deziel, Chris. How To Find Slope From An Equation last modified March 24, 2022. https://www.sciencing.com/how-to-find-slope-from-an-equation-13712210/