Draw a line on a coordinate plane.
The line you just drew has a very simple description as an equation in two variables.
That is, there's a very simple equation in $\,x\,$ and $\,y\,$ that will be true
for every point on the line,
and false for every point that's not on the line.
Here's a preview of coming attractions:
If the line isn't vertical (straight up/down), then the equation of the line will look like this:
$$
y = (\overset{\text{we'll call this }\,m}{\overbrace{\text{some number}}})x +
(\overset{\text{we'll call this }\,b}{\overbrace{\text{some number}}})
$$
With the names $\,m\,$ and $\,b\,$ in place, the equation of the line takes this form:
$$
y = mx + b
$$
Here are some examples of equations of this form:
$y = 2x + 3$ | ($m\,$ is $\,2\,$ and $\,b\,$ is $\,3$) |
$y = 3x + 2$ | ($m = 3\,$ and $\,b = 2$) |
$y = \frac13x - 7$ | ($m = \frac13\,$ and $\,b = -7$) |
$y = 5$ | (rewrite as $\,y = 0x + 5\,$ to see that $m = 0\,$ and $\,b = 5$) |
It ends up that the coefficient of $\,x\,$ in this equation (which we've called $\,m\,$) gives information
about the slant of the line.
That is, the number $\,m\,$ (which might equal zero) will answer questions like this:
The purpose of this section is to begin to develop your intuition about the
slope of a line.
The next section, Practice with Slope, will make the ideas more precise.
In the web exercise that follows these examples, you'll be doing the computations
and filling in the information that is highlighted in
green.