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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.
Slope-Intercept
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 of the line and the y-intercept are easily identifiable. The slope is the steepness of the line, and the y-intercept is the place the line crosses the y-axis.
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 a thing (often called x), if you add one more you always get a consistent amount more of the other thing (often called y).
Let’s look at some examples of linear relationships:
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.
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.
It’s important to know that all relationships are not linear. For example, the number of dogs you have and the amount of dog poop you have to clean up in the backyard is not linear. Some dogs make bigger messes than others.
Why Linear Relationships Are Important
This investigation of linear relationships has a purpose: to help you understand that a line (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 * 400). If you eat 3 slices, you get 1200 more calories (3 * 400). 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 = 400x
This is just the algebraic way of writing: Calories = 400 * Number of slices
Just by glancing at the equation y = 400x you can tell the slope of the line is 400. It is the amount y increases if x increases by one.
 
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