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Lines and Slope: Chameleon Graphing

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Using Slope

We can use slope to help us compare lines. We already know what it means when one line's slope is greater or smaller than another line's slope: the line with greater slope has a steeper slant.

What if two lines have exactly the same slope? They must slant exactly the same amount. This doesn't mean that the two lines are identical. For instance, in this picture the green line and the straight part of Joan's tongue both have a slope of 1.

two lines with slope 1

Will Joan's tongue ever touch the green line? You might use the picture to guess that it won't. Joan's tongue and the green line are not touching now. If Joan stretches her tongue out farther without changing its slant, her tongue and the line shouldn't ever get any closer together.

If two lines never meet, we say that they are parallel. Parallel lines always have the same slope. Also, if two different lines have the same slope, they are parallel. (And, obviously, a line always has the same slope as itself.) Since Joan's tongue and the green line have the same slope, they really won't ever run into each other. The guess we made from the picture was right.

Sometimes it's difficult to tell whether two lines are parallel from just a picture. Do you think these two lines will ever meet?

two lines

Let's find the slope of the lines to check whether or not they are parallel. We'll start with the slope of the blue line:

    m = (y2 - y1) / (x2 - x1)
    = (4.3 - 2.1) / (3 - 1)
    = 2.2 / 2
    = 1.1

Now let's find the slope of the red line:

    m = (y2 - y1) / (x2 - x1)
    = (3 - 1) / (3 - 1)
    = 2 / 2
    = 1

The two lines have different slopes, so they cannot be parallel. They must run into each other somewhere.


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to Ursula Whitcher

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