Slope-Intercept Form: Definition & Examples

Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

The slope-intercept form of the equation of a line is a useful form for graphing as well as for understanding the relationship between x and y. In this lesson, learn how the slope-intercept form helps you understand the equation of a line.


The equation of a line can be written many different ways, and each of these ways is valid. The slope-intercept form of a line is a way of writing the equation of a line so that the slope, or steepness, of the line as well as the y-intercept, the place the line crosses the vertical y-axis, are easily identifiable.

Lines and Linear Relationships

A line is a relationship between two things - but not just any relationship. When you have a linear relationship, one that can be graphed as a line, there is one big condition:

No matter how much you have of one of those things (often called x), if you add one more you always get a consistent amount more of the other thing (often called y).

Some linear relationships:

1. The amount of pie you eat and the number of calories you consume. If each slice of pie has 400 calories, and you eat one more piece, you will have consumed 400 more calories. It is totally irrelevant how many pieces you have already eaten.

2. The number of steps you take (of consistent size) and the distance you travel. If you take 100 more steps, and you can travel 1.5 feet with each step, then you have traveled 150 more feet, regardless of how far you've already walked.

Some relationships that are not linear:

1. The number of dogs you have and the amount of dog poop you have to clean up in the backyard. Some dogs make bigger messes than others.

2. The width of a square bedroom and the area inside it. If you make a 3 meter room bigger by 1 meter, you have added 7 square meters. But if you make a 5 meter room bigger by 1 foot, you have added 11 square meters, as shown in the picture.

Increasing the width of a shape does not linearly increase the area.
Two rooms, one small and one large

Why Linear Relationships Are Important

This investigation of linear relationships has a purpose - to help you understand that a line, or linear relationship, always suggests that increasing x a certain amount has a constant effect on y.

Let's return to the pie example. Every time you eat one more slice, you get 400 calories (assuming all the slices are the same size). So, if you eat two slices, you get 800 more calories (2 x 400). If you eat 3 slices, you get 1200 more calories (3 x 800). This amount more you get if you have 1 more of something when a relationship is linear is called the 'slope.'

For our pie example, pretend for a moment that pie was all you ate for dinner. Call the number of slices you ate x. If you ate 2 slices, x = 2. If you ate 9 slices, x = 9. In each case, the number of calories you ate is y. How do you get from pie slices to calories? You multiply, like this:

y = 400 x

This is just the algebraic way of writing:

Calories = 400 x Number of slices

Just by glancing at the equation y = 400 x you can tell that the slope of the line is '400.' It is the amount y increases if x increases by one.

Now let's pretend that you did not just eat pie for dinner. Maybe you had roast beef and potatoes, totaling 640 calories. You have to add that 640 calories to your total calorie consumption. Your total calories is still y and your number of pie slices is still x. You aren't really sure how many slices you will eat yet. But no matter how much pie you eat, you will have to add 640 calories to it. By adding '640' to the previous equation you get:

y = 400 x + 640

You can still tell how many calories each slice of pie will add: 400. But, the 640 tells you something extra. It tells you how many calories you will consume if you eat NO pie. It might seem obvious - after all, you had 640 calories before you got to dessert. The equation just formalizes the relationship mathematically. If you put a '0' in for x, you get y = 640. No pie means no additional calories. This y-value that you get if x=0 is called the y-intercept.

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