Linear Equations ....................................

Slope
The slope of a line is a measure of how steep it is. There are a number of ways of measuring the steepness of a line. For example, highway steepness, or 'grade', is measured as a percent. In trigonometry, steepness is the tangent of the angle of inclination of the road.

We'll use the more general method to measure steepness, which is 'rise divided by run'. This method lends itself nicely to an algebraic approach, which will be necessary later.

We'll use the symbol m for slope.

Here is a line that, left to right, is moving upward. It passes through points A and B.
If you count squares, you can see that, from A to B, you move horizontally a distance of 8, and then rise vertically a distance of 6.

The slope of the line is rise divided by run, or 6 divided by 8, which is 0.75, as shown above.

If you think about it:

A line that rises more than this one, in the same distance, will have a bigger slope.

A line that does not rise as much as this one, in the same distance, will have a smaller slope.

A line that rises the same amount that it runs will have a slope of exactly 1, since rise = run.

Now let's look at what happens when a line slants downwards.

Here is a line that, left to right, is moving downward. It passes through points A and B.
If you count squares, you can see that, from A to B, you first drop vertically a distance of - 6, and then travel a distance of 8 horizontally.

The slope of the line is rise divided by run, or -6 divided by 8, which is -0.75, as shown above.

If you think about it: