Today, we’re going to look at slope — the grade of a straight line on the coordinate plane. Specifically, we’re looking at how to use the rise over run formula to measure a line’s slope. This is sometimes also called the “slope formula,” which states that slope = (change in y)/(change in x), or (y1 − y2)/(x1 − x2). (Many of Maghoosh’s test prep students ask us about slope formula and rise over run, especially as it pertains to Magoosh GMAT coordinate geometry and Magoosh GRE coordinate geometry.)
In the Slope Formula, what is Rise Over Run?
As you consider the slope formula, remember that there are two ways to talk about slope: graphically and numerically. “Rise” and “run” are just ways of thinking about how we could generate a line. For example, if we have a slope of 2 and we’re on one point, we would move up 2 (rise) and to the right 1 (run) to find the next point. Take (0,0) as our starting place, which is the red point in the picture below. Up 2 (in the positive vertical direction) and to the right 1 (in the positive horizontal direction) would be the point (1,2). We can keep going. We could go up 2 more and to the right 1 more and we get to (2,4). Then, if we repeat this process again and we get to (3,6), and then (4,8), and so on.
Each point is up 2 and 1 to the right from the previous point. With a positive slope like 2, we can also go down 2 and to the left 1. It’s really just doing the up 2 and to the right 1 in reverse. If we started at (4,8), down 2 and to the left 1 would be (3,6), then (2,4), then (1,2). Finally, we’re back at (0,0), and we could keep going. (–1,–2), (–2,–4), and so on.
Finding Points and Building the Line With the Rise Over Run
Many problems rely on this approach — we use the slope formula to generate points on the line. Of course, lines are a collection of ALL points that are related by a given slope. You can see it in the image above. This is the line defined by the equation y = 2x, and if we graphed y = 2x we’d fill in all the gaps to make a straight line:
We can use the rise/run method to generate those points too! Remember we want a rise of 2 and a run of 1, but we don’t have to step by 2 and 1 to generate points. What about a rise of 1 and a run of ½? 1 divided by ½ is 2, just like 2 divided by 1. Start from (0,0) just like we did before. Next, go up by 1 and right by ½:
Decimals or Exponentially Large Numbers in the Rise Over Run
What about a rise over run that’s much smaller? How will rise over run be applied to the slope formula in that case?
0.1/0.05 = 2, so it’s still equivalent to a slope of 2.
It should be clear at this point that as long as the ratio of the y vertical component and the x horizontal component is equal to 2 in the slope formula, then the point will still be on the line. We can take as small a step in the slope formula as we want:
And as large a rise over run step as we want:
If the step in the vertical direction we take is twice the step in the horizontal direction — if the ratio of rise/run is 2/1, then the point will be on the line. That’s it! Whatever slope you get in a problem, you can use rise/run to determine where another point on the line will be.
A Special Note on Fractions and How they Impact the Slope Formula
One final point before we close. Fraction slopes have an added complexity, even though they’re really not any different. Certainly, if we have a slope of –2/3, you can read the rise and run right off the fraction. We go down 2 and right 3 or up 2 and left 3.
Remember that in the slope formula, up and right are positive and down and left are negative. That means that when we combine those actions, up+right and down+left are positive and up+left and down+right are negative. That’s because for up and right, rise over run involves dividing a positive number by a positive number, which gives a positive slope. If it were down and left, then rise over run would be a negative number divided by a negative number, which also gives a positive slope. If we went down and right, the slope formula would involve a negative number divided by a positive number which gives a negative slope. Finally, up and left would be a positive number divided by a negative number, so the slope would be negative.
But what about all the points in between? Lines are a collection of every point, not just integer coordinates. Instead of going from (0,0) to (3,–2), we could go to a point that has an x-coordinate of 1. What would the y-coordinate be? It would be –2/3. In this iteration of rise over run, you can imagine stepping to the right one more, where the x-coordinate is 2. There the y-coordinate is –4/3. Go to the right one more to x=3, and the y-coordinate is –6/3, or –2. That’s it! That’s what we did when we went down 2 and right 3, but we did the slope formula in steps.
The Takeaway for Slope Formula and Rise over Run
That’s how the slope formula works! You can take large steps or tiny steps along a line, as long as the steps are proportional to each other. We determine that proportion by the slope, rise over run.