Be the Professor: All About Slope

Hello! I'm Lucas, and I'm here to teach you all about slope. This includes all about how its used, how it works, how it can help you solve all types of problems, and real-world applications for slope and how they apply to what you are learning today.

First I will start out by explaining the basics you need to understand to deepen your understanding of the slope of a line. The general equation for any straight line is Ax+By+C=0. A or B can be zero, but both cannot be zero at the same time. There are also two other forms, and those are point-slope form, which is y-y1=m(x-x1), and the other is slope-intercept, (the most important one for this lesson), and that is y=mx+b.

In slope-intercept form, (which is the most important one for this lesson, remember?) "b" is the y-intercept and "m" is the slope, and "x" and "y" are obviously two points on the line. Knowing these two alone can help you to read a graph of a straight line easily as well as construct the slope-intercept formula.

You can get the equation for any straight line in any form you want to by using algebra, to either get x and y on the same side for standard form, y alone for slope-intercept, or y and x on different sides for point-slope form. The definition of slope (in simple terms) is how steep a straight line is. Another way to think about slope (in a less simple but still pretty simple way) is to think about the phrase "rise over run"(rise/run), meaning you use two points on the line and find how far the second point is from the first point vertically, then horizontally, hence the whole "rise over run" saying. So using the two points on the graph is how to find the slope of a line is only one of the ways that slope can be found.  I have included a diagram below to show how the "rise over run" saying is used to find the slope of two points on a graph.

The next way to find the slope of a line is to use either two points that are given in a problem or given on a graph. Using these two sets of points, you can plug these points right into the (very convenient) slope formula, which is shown in the big picture below...

"m" is always the slope, and the y's and x's in that are the points that are given from either the graph or the problem.

Although slope may seem like another one of those useless math skills that you may think you will never use in real life, there are actually some  real life situations that you wouldn't think require slope skills. Some common applications of slope is building roads, and figuring how steep they have to be, constructing ramps for wheel chair ramps, building staircases, and also in many different sporting events where ramps are needed, such as motocross, biking, skiing, snowboarding, and plenty others.

So all in all, slope is a pretty simple concept that will help you in numerous different aspects in your mathematics career. It is a pretty easy formula to remember, and the phrase "rise over run" is sort of catchy and pretty hard to forget whenever you hear the word 'slope'. I hope this lesson has helped and if you want to solidify that you've grasped the concepts of slopes, then I highly recommend that you do some of the practice problems that I have included below to totally crush that test you have coming up on slope!

1.) Find the slope of the line passing through the points (4,3) and (-5,-2)

2.) Find the equation of the line that passes through the points (3,4) and (-4,6)

3.) Find the slope of the line passing through (2,1) and (12,18)

4.) Find the slope of the line on the graph below

5.) What is the y-intercept and slope of the equation y=-2x+6?