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*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.*

We also recommend watching Abstract Algebraic Examples and Going from a Graph to a Rule and

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

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*).

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.

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.

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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The equation *y* = 400 *x* + 640 is in slope-intercept form because we can look at it and immediately see the slope and the *y*-intercept. Usually, this form is described as ** y = m x + b** form. 'm' is the slope and 'b' is the

This equation is *not* the only *correct* way to write the equation of this line. For example, you could do a little algebra and rewrite it like this: -400 *x* + *y* = 640. This form is still the same relationship between *x* and *y*. It is still the same line. If you put '0' in for *x*, you still get 640 for *y*, but it is *not* in slope-intercept form.

Graphing is very simple if the line is written as *y* = m *x* + b. First, you identify the y-intercept, the place where the line crosses the *y*-axis (the thicker vertical line) and draw it as a point on the graph.

For example, suppose you have the line *y* = 3 *x* + 1. The *y*-intercept is at *y* = 1, as shown in the picture.

Next, you use the slope to help you draw the correct steepness of the line. In this case, the slope is '3.' So, *y* increases by 3 every time *x* increases by 1. From the *y*-intercept, move in the *x*-direction 1 unit, go up 3 units and make a second point, as shown in the picture.

Once you have two points, drawing in the line is easy.

While the slope-intercept form of a line isn't the *only* way an equation of a line can be written, you will probably find that it is easy to understand and easy to graph - and a very useful way to understand the relationship between many things.

The slope-intercept form is simply the way of writing the equation of a line so that the slope (steepness) and *y*-intercept (where the line crosses the vertical *y*-axis) are immediately apparent. Often, this form is called ' *y* = m*x* + b' form. This form of a line is one of the most commonly used in algebra classes.

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