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).