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# Slope and Intercept Study Guide

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Updated on Oct 3, 2011

## Introduction to Slope and Intercept

The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality.

—Cletus O. Oakley (1909–1990) Mathematics Professor

In this lesson, you'll learn about the slope and y-intercept of the equation of a line, how to put an equation in slope-intercept form, and how to find the slope and y-intercept of a line.

The variables x and y can be used to form the equation of a line. A horizontal line takes the form y = c, where c is any constant, such as 0, 3, or –4. A vertical line takes the form x = c, such as x = 2 or x = –14. The equations of all other lines contain both x and y. These lines have a slope and a y -intercept. Slope is the change in the y values between two points on a line divided by the change in the x values of those points. The y-intercept of a line is the y value of the point where the line crosses the y-axis.

We write the equation of a line in slope-intercept form so that it is easy to spot the slope and y-intercept just by looking at the equation. Slope-intercept form is y = mx +b, where m is the slope of the line and b is the y -intercept.

#### Example

y = 2x + 5

This equation is already in slope-intercept form. We can tell that it is in slope-intercept form because y is alone on one side of the equal sign, and the other side of the equation contains no more than two terms, with x appearing in no more than one term. The coefficient of x is 2, which means that the slope of the line is 2. The constant that is added to 2x is 5, which means that 5 is the y-intercept of the line.

If no constant is added to the x term, then the y-intercept of the line is 0. The equations y = 4x, y = –3x, and all have a y-intercept of 0.

It is also possible for a line to have a slope of 0. If x does not appear in the equation of a line, then the line has a slope of 0. The lines y = 4, y = –10, and y = 0 are all in slope-intercept form, and they all have slopes of 0.

#### Tip:

Vertical lines—lines in the form x = c—do not have a slope of 0. They have no slope at all. These equations are not in slope-intercept form, and they cannot be put in slope-intercept form because they do not contain the variable y.

## Putting an Equation in Slope-Intercept Form

If the equation of a line is not in slope-intercept form, we must perform one or more operations to get y alone on one side of the equation, with x in no more than one term on the other side of the equation.

The equation y + 9 = 3x is not in slope-intercept form, because the constant, 9, is on the same side of the equation as y. To remove 9 from the left side of the equation, subtract 9 from both sides of the equation: y + 9 –9 = 3x –9, and y = 3x –9. The equation is now in slope-intercept form, and we can see that the slope is 3 and the y-intercept is –9.

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