The slope of a linear equation is a number that tells how steeply the line on our graph is climbing up or down. If we pretend the line is a mountain, it's like we're talking about the slope of a mountain. If it helps you, draw a snowcap at the top. Some mountain climbers. A ski lift. Nothing too elaborate though.
We move from left to right on the x-axis, the same way that we read. If the line gets higher as we move right, then we're climbing the mountain, so the line has a positive slope
If the line gets lower as we move right, then we're descending the mountain so the line has a negative slope.
If we stay at the same height, then the slope is zero because we're not going up and we're not going down. Pretty boring mountain, if you ask us.
Now let's find some actual numbers for slopes. Thinking of the mountains, a slope is a ratio that describes how quickly our height changes as we move over to the right. Not our actual physical height, mind you. We won't be getting shorter or taller throughout the course of these examples, even if you do feel by the end of it that you've grown.
Julie is climbing a mountain. For every 10 feet Julie travels (measured along the ground), she goes 20 feet higher. What is the slope of the mountain?
The slope of the mountain is .
For every foot Julie travels (measured along the ground), she gets 2 feet higher off the ground. She'd be even higher off the ground if she'd worn heels, but we suppose those would have been an odd choice for mountain climbing.
Check this out. Look at the graph of the line y = x:
The slope of the line y = x is 1. If we move over to the right by 1 on the x-axis, we also move up by one on the y-axis:
Find the slope of the line pictured below. If we haven't heard from you in three hours, we'll send the park ranger after you.
Let's look at what happens between a couple points of the graph:
On this line, or mountain, we move up 2 for every 3 we move over. Except for that one time we moved up 2, encountered a mountain lion, and ran back down 7.
Our slope is .
One way to think about slope is
To use this formula to find the slope of a line, we first fix two points on the graph whose coordinates we can easily figure out.
The rise is the amount y changes between those two points, and this number may be positive or negative. Remember, you can be going up or down the mountain. The run is the amount x changes between those two points. We usually think of moving from the point on the left to the point on the right, meaning that x is increasing and the "run'' is always positive. You might climb up or down, but you would never run backwards, right? Aside from when you were backing away from that mountain lion, we mean.
Then the slope of this line is
Be careful: It's all very well and good to memorize the formula
but in order to use it correctly, you need to know what "rise'' and "run'' really mean. It doesn't refer to your underwear rising up on you or your stockings having a run in them, although either would be a wonderfully memorable image. It might be helpful to have a picture like the below in your head, in addition to the formula:
Find the slope of the line shown below.
Let's find a couple of points whose coordinates are nice and easy to work with, and see what the rise and run are between those two points. Use the undergarment visual if you'd like. It won't help you with this problem, but no one's stopping you.
The slope is
If we try to apply this formula to a vertical line, we'll be in trouble. Since the "run'' between any two points on a vertical line is 0, and we can't divide by 0, the slope of a vertical line is undefined. Makes sense, since it would take some powerful thighs to run directly up a vertical mountain. If you attempted to do so, you'd soon be undefined as well.
If you know what we mean.
The slope of the line x = 1 is undefined:
We can also find the slope of a line if given any two points on the line.
Find the slope of the line that passes through the points (1, 3) and (2, 7).
We know part of the line will look like this:
To get from the point (1, 3) to the point (2, 7), we need to move right 1 and up 4:
That means the slope of the line is . Yodelay-hee-hoo!
Find the slope of the line that goes through (-3, 1) and (2, -2).
Part of the line looks like this:
The distance we travel to get from one value of x to the other is 3 + 2 = 5, since first we have to travel from x = -3 to x = 0 and then from x = 0 to x = 3. We have a layover at the y-axis, where we can grab a quick bite of vastly overpriced fast food while we wait for our connecting line.
To get from one value of y to the other, first we travel from y = 1 to y = 0, then from y = 0 to y = -2, for a total rise of -3. Once again, we couldn't get a direct flight. Ah, well. It'll give us more time to read this book we've been working on.
Thus the slope of this line is .
Be careful: It's common to make mistakes calculating the rise and run when there are negative coordinates involved. To avoid mistakes, we recommend drawing a picture to help with the calculations. If you're worthless as an artist, find a mountain or book a flight so you can live out one of our previous examples. More expensive and time-consuming to get the point across that way, but it will certainly drive the idea home.
Well, now we can read off the slope of a line from a graph or from any two points on the line. We're feeling good about ourselves. How about graphing a line if given a single point and a slope?
Graph the line that goes through (0, 0) and has a slope of 2.
Let's start by drawing the point we're given:
We're told the line has a slope of 2, which means as x moves over 1, y goes up 2:
We now have two points, which is enough to draw a line: