Lines are often more complicated than simple vertical and
horizontal lines as we looked at in the last section. The
slopes of the lines we worked with so far were 0(perfectly horizontal) or
undefined(perfectly vertical.) However, there are an infinite number of
slopes in between these two extremes. We can find the slope of the line
between two points, and , with
the following equation:

For example, take the points (-2,1) and (5,4). We can see their location and the slope of the line connecting them.

The rise of the line from (-2,1) to (5,4) is 4 - 1 = 3. The run is equal to 5 - -2 = 7. Putting the two differences together we have 3/7. The slope of the line is equal to 3/7. For every 7 units the line goes across, the line goes up 3 units.

Slope and y-intercept may
be found by manipulating an equation of a
line
into the form

Here m is the slope and b is the y-intercept. Let's try it on an equation:

We need to isolate the y. Lets do this by adding 10x to both sides of the equation which gives us:

The line is not quite in point-slope form. We can divide the entire equation by 2 which reslults in:

Now we can read off the slope and y-intercept. The slope of the line is 5 = 5/1. That means for every one unit in distance the line goes to the right, the line will rise 5 units. b = 3 that means the line crosses the y-axis at 3. The line is shown in red in the graph below.

The graph also shows two extremes in slopes. The green horizontal line is perfectly flat. The equation of that line is y = 3. Since there is no x term, the coefficient is understood to be 0. We can represent it with y = 0 * x + 3. The slope of the line is 0.

The yellow vertical line is represented by the equation x = -2. This line is not in the form y = mx + b because there is no y term. The slope on this line is undefined since the run of the line is 0, but the rise of the line is infinite.