In Mathematics, a slope is a way of explaining a line. This article will explain real life slopes, compare them to algebraic slopes, and finally, explain why algebraic slopes are important.
A real life slope is what you will find at the base of a hill. The ground starts out flat and slowly rises the closer you get to the top. If you follow the ground as it continues to rise, you will find there is a slope the entire way to the top. Sometimes the ground may be flat, but other time the ground may be steep. Think about what makes the slope flat or steep. If you walk a long distance, and your elevation doesn’t change, the slope must be flat. If you walk a short distance, and your elevation changes a lot, then the slope must be steep.
This idea of a real life slope is the exact same as an algebraic slope. Imagine you have a line graphed on the Cartesian plane. If you want to know the slope of that line, you need to know two things. First, what is the elevation change between two points on the line, and second, what is the change in distance between those same two points. Elevation is described by the y coordinates on the Cartesian plane. Distance is described by the x coordinates.
Here is an example. A line goes through the points (1,2) and (7,3). In order to find the slope, you need to find the elevation change. Look at the y values first. The starting elevation is 2 and the final elevation is 3. Subtract the starting elevation from the final elevation (3-2), and you will get an elevation change of 1. Now look at the x values to find the change in distance. You started at 1 and ended at 7. Final distance minus starting distance (7-1) gives you a change in distance of 6.
Slope is mathematically defined as rise (elevation change) over run (distance change). So the slope is 1/6.
The reason this number is important, is because it allows you to simplify a problem. If you were working with a bigger problem, you would no longer have to worry about distance and elevation change, because you have combined these two measurements into a single number called slope. A great deal of mathematics deals with this same sort of problem simplification. It allows physicists to solve complicated problems like finding the distance between two planets, or calculating the force of gravity on the moon.
In conclusion, both an algebraic and real slope have a change in elevation and a change in distance. An algebraic slope is defined as the rise (the elevation change between two points on a line) over the run (the change in distance between those same two points). By expressing elevation and distance change in a single number called slope, you can simplify a problem and make it easier to solve.