In this lesson, we'll discover all about the slopes of lines. We'll learn about different types of slopes and how to find the slopes of lines. We'll use two methods: the slope formula and the slope-intercept form.
What Is a Slope?
How often do you think about slopes? Maybe not that often. But they're pretty important. I used to live in San Francisco where I'd ride my bike around. Slopes are very important there. They're kind of unavoidable. The greater, or steeper, the slope, the less likely I wanted to take that route. I wanted to find the flattest route, or the route with the smallest slopes.
When you're working with lines, that's all slopes are - they're like different types of hills. Fortunately, you don't have to try to pedal your bike up the slopes of lines. Officially, slope is defined as the steepness of a line. We use the letter m to signify a line's slope.
Why m for slope? There isn't even an m in the word slope. Some people say it's because the French verb for 'to climb' is 'monter,' but that may be a myth. I like to think that it's because the letter M is a slope-filled adventure. Notice how it has slopes built right in! But I made that up.
So how do you find a line's slope? You might hear slope referred to as 'rise over run.' That's a way of saying that a slope is a line's vertical change divided by its horizontal change. Or, how much it rises over how much it runs.
The formula for finding a slope is m = (change)y/(change)x, which equals (ysub2 - ysub1)/(xsub2 - xsub1). Those ysub1s and xsub1s just symbolize points on the line. To find a slope, you need two points on the line: (xsub1, ysub1) and (xsub2, ysub2). Since we're talking about straight lines, the slope is constant, so it doesn't matter where those points are. Knowing how the line changes vertically (or its rise) over how it changes horizontally (or its run) is all we need.
Sometimes, you'll be given two points on a line and asked to find the slope. If so, just plug those points into the formula. Just remember: rise over run means the y value goes on top, so the y-axis is the vertical one.
Types of Slopes
Before we try out the slope formula, let's look at what a slope really is. Here's a line with a slope of 1/2. That means it goes up 1 space for every 2 it goes over. That's its rise over run: 1 over 2.
Here's one with a slope of 2. If you remember your fractions, you know that 2 is just 2/1, so this line goes up 2 for every 1 it runs.
Those slopes were positive. What about negatives? Rise over run doesn't change. We just go in the other direction. Here's a line with a slope of -2/3. It still rises 2 for every 3 it runs. It just runs the other way.
Just remember that slopes work like so many things that normally go left to right - everything from words on a page (at least in English) to the time bar on this video. So, positive slopes rise from left to right. Meanwhile, negative slopes rise from right to left, or backwards.
Oh, and there are two other kinds of slopes. Check out this line. If you were riding on this hill, well, that'd be all right. Horizontal lines have a slope of 0. Why? Well, it's not rising, so the numerator on the fraction is 0. 0 divided by anything is 0.
Then there's this one. I think I remember a street in San Francisco that looked like this. As for the slope, it will have a 0 on the bottom of the fraction. What happens when you divide by 0? Your calculator gets angry. Vertical lines have undefined slopes.
Practice Using Points
Let's try finding a few slopes using the slope formula. Let's say we're told a line goes through (1, 2) and (3, 3). What is the slope? Remember the formula? m = (ysub2 - ysub1)/(xsub2 - xsub1). What are our y values? 2 and 3. And our x values? 1 and 3. Let's plug them in. We get m = (3 - 2)/(3 - 1). That's 1/2. So our slope is 1/2. In the graph, you can see it rises 1 and runs 2.
Rise and run - it kind of sounds like a zombie movie, doesn't it? And like in a zombie movie, you wouldn't run until they rise, right? It's always rise first, then run. Or, you know, stay and fight. Me? I'm running.
Ok, here's another. We have a line passing through (-1, 3) and (-4, -2). Don't lose track of those negatives. Let's plug it into our formula. m = (-2 - 3)/(-4 - (-1)). That's -5/-3, or 5/3. So it rises 5 and runs 3. Note that despite all the negatives, this is still a positive slope.
Let's try one more of these. We have two points: (1, -2) and (-3, 1). Using our formula, we get m = (1 - (-2))/(-3 - 1). That's 3/-4, or -3/4. A negative slope! And it looks like this. Yep, that's negative.
I should note that it doesn't matter what you make ysub1 and ysub2. If we try that last one, reversing the order of the points, we get m = (-2 - 1)/(1 - (-3)), which is -3/4. The same thing! So don't worry about order. Just focus on rise over run.
Slopes from Equations
What do you do if you're not given two points? What if, instead, you have an equation of a line? Maybe you need to find the slope of 2x + 3y = 12. Well, you could try to find two points on the line, then use the slope formula. But there's an easier way.
You just need to move things around so your line is in slope-intercept form. Slope-intercept form looks like this: y = mx + b. As before, m is the slope. b is the y-intercept. That's why it's called slope-intercept form. Why is b the y-intercept? Well, remember that the y-intercept is where x = 0. If we plug 0 in for x in y = mx + b, we get the y-intercept.
So in 2x + 3y = 12, just get y alone on one side of the equation. First, subtract 2x from both sides to get 3y = -2x + 12. Then divide both sides by 3. We get y = -2/3x + 4. Remember, the slope is the coefficient of x. So the slope of this line is -2/3.
If you come across an equation that already looks like y = mx + b, like y = 3/4x - 5, then your work is done for you! Whatever is in front of that x is your slope.
Practice Using Equations
Let's practice converting a few equations. Here's one: y - 4x = 2. Remember, we need to make it look like y = mx + b. So just add 4x to both sides to get y = 4x + 2. That's it! Our slope is 4, right here!
What about this one? x + 4y = -12. Ok, let's first subtract x from both sides. 4y = -x - 12. Note that I could've done -12 - x, but I want my x listed first to make sure I'm matching the slope-intercept form. Next, let's get that y alone by dividing both sides by 4. We get y = -1/4x - 3. Our slope? Right here. -1/4.
Ok, how about one more? 5x - 4y = 0. Wait. Do you see what's different? How can our b be 0? Remember, the b in y = mx + b is the y-intercept. The y-intercept is the place where x = 0. So this line crosses the x- and y-intercepts at the same time, at (0, 0). But what about the slope? Let's subtract 5x from both sides to get -4y = -5x + 0 (we don't really need that +0, but let's keep it as we practice the slope-intercept form). Next, divide by -4. We get y = 5/4x + 0. And our slope? 5/4. And guess what? Now we even know two points on this line: (0, 0) and (4, 5).
In summary, slope is just the steepness of a line, and it's represented by m. A positive slope goes up from left to right, while a negative slope goes up from right to left. The slope of a horizontal line is 0, while a vertical line has an undefined slope. We can find the slope of a line when we're given two points using the slope formula: m = (change)y/(change)x, or (ysub2 - ysub1)/(xsub2 - xsub1). That's just rise over run. If we're given an equation, just rearrange it to the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. And if you're ever in San Francisco, watch out for those vertical streets. They're dangerous!
At the end of this lesson, you should be able to:
- Define slope and recall slope-intercept form
- Solve problems asking you to find the slope of a line