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 Dependent Variable

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# Slope

Objective Learn the concept of the slope of a line and to evaluate it for particular lines.

In this lesson, you will be introduced to the notion of the slope of a straight line. It is very important that you see many examples, and that you do many exercises by yourself. In these exercises, you should use the formula for slope so that you become familiar with using it. You should also write and solve your own exercises involving slope. It is desirable that you have grid paper for this lesson.

## Slope

First, let's talk in intuitive terms about what is meant by slope. We can assign a number that allows us to measure the steepness of a straight line. Also, the greater the absolute value of this number, the steeper the line will be. If you draw a straight line with two points on it, A and B, there are two numbers attached to this pair of points, namely the rise and the run . The rise is how much higher B is than A in the vertical direction, and the run is how far over from A point B is in the horizontal direction.

Definition of Slope

Words The slope is the value of the quotient .

Model

So far, we have talked about lines without placing them in the coordinate plane. Let's try to understand slope better by studying lines in the coordinate plane. Start by plotting the sequence of points (1, 1), (2, 2), (3, 3), and (4, 4) and try to find out what the pattern is. Find the value of y in ( -1, y ). Next, plot (1, 2), (2, 4), (3, 6), and (4, 8) on another coordinate plane and try to find out what the pattern is. What do the two sets of plotted points have in common? (Both sets of points lie on a straight line.)

The two points plotted at the figure above have coordinates (1, 2) and (2, 4). The rise is the difference between 4 and 2, so it equals 2. The run is the difference between 2 and 1, so it equals 1. The slope is the quotient or 2. Next, we choose two different points on the same line, say (0, 0) and (3, 6), and compute the slope again.

Key Idea

The slope is the same for any pair of points on the same straight line. Therefore, it is not necessary to refer to a particular pair of points when speaking of the slope of a line.

Note that the y-difference is the numerator and the x-difference is the denominator.