Algebra Basics: Solving 2-Step Equations - Math Antics

Hi, I'm Rob. Welcome to Math Antics. In the last two Algebra videos, we learned how to solve simple equations that had only one arithmetic operation in them. But often, equations have many different operations, which makes solving them a little more complicated. In this video, we're going to learn how to solve equations that have just two math operations in them – one addition or subtraction, and one multiplication or division. And the concepts you learn in this video will help you solve even more complicated equations in the future. Now, as you might expect, equations that have two arithmetic operations in them are going to require two different steps to solve them. In other words, to get the unknown all by itself, you'll need to undo two operations. But that doesn't sound too hard, right? I mean, we learned how to undo any arithmetic operation in the last two videos. And that's true, but there's a couple reasons that make two-step equations a little trickier to solve. The first is that there's a lot more possible combinations of those two operations. And the second is that when there's more than one operation, you have to decide what order to undo those operations in. Uh… hello? If you need to know what order to do operations in, just follow the order of operations rules. You did watch that video, didn't you? I sure did. But, the order of operations rules tells us what order to do operations, not what order to undo them. Uh… well, then could we reverse the order since we're undoing the operations? Now that's a good idea! Well, of course it is! When solving multi-step equations, that's basically what we're going to do. Using the order of operations rules in reverse can help us know what order to undo operations in. But it can be a little tricky actually putting it into practice. So, to see how it works, let's start by solving a very simple two-step equation. 2x plus 2 equals 8. In this equation, the unknown value, x, is involved in two different operations, addition and multiplication, which is implied between the first two and the x. And to undo those two operations, we need to use their inverse operations, subtraction and division. But the question is, which one should we do first? Like many things in life, the order we decide to do things in can make a big difference. Oh come on! There's gotta be an easier way! First socks, then shoes! Fortunately, in math, we have a special set of rules that tell us what order to do operations in. Those rules tell us to do operations inside parentheses or other groups first. And then we do exponents, and then we do multiplication and division, and last of all, we do addition and subtraction. Those are the rules you need to follow when simplifying mathematical expressions or equations. But solving an equation is different because we're trying to undo any operations that the unknown value is involved with so that the unknown value will be all by itself. So when solving equations, the best strategy is to apply those order of operations rules in reverse. Using the reverse order of operations is not the only way to solve a multi-step equation, but it's usually the easiest way. Just like it's much easier to take your shoes and socks off in the reverse order that you put them on. Gah! Are you sure it's socks before shoes? Since the order of operations rules tell us to do multiplication BEFORE we do addition, we should undo addition BEFORE we undo multiplication. So first, we undo the addition by subtracting 2 from both sides of the equation. On the first side, the plus 2 and the minus 2 cancel each other out, leaving just 2x on that side. And on the other side, we have 8 minus 2 which is 6. Next, we can undo the multiplication by dividing both sides of the equation by 2. On the first side, the 2's cancel, leaving x all by itself. And on the other side, we have 6 divided by 2 which is just 3. There, we've solved the equation using the order of operations rules in reverse, and now we know that x equals 3. That wasn't so bad, was it? Let's try solving another simple two-step equation that has division and subtraction in it. x over 2 minus 1 equals 4. Again, we're going to apply the order of operations rules in reverse to undo the subtraction and the division operations. Since we would normally do the subtraction last, we're going to undo it first. To undo the subtraction, we add 1 to both sides of the equation. On the first side, the minus 1 and the plus 1 cancel out, leaving just x over 2 on that side. And on the other side, we have 4 plus 1 which is 5. And then, to undo the divided by 2, we need to multiply both sides by 2. On the first side, the 2's cancel, leaving x all by itself. And on the other side, we have 2 times 5 which is 10. So our answer is x equals 10. Those examples are pretty easy, right? But solving two-step equations gets a bit trickier thanks to a little something in math called groups. Do you remember how parentheses are used to group things in math? And our order of operations rules say that we're supposed to do any operations that are inside parentheses first. In other words, we need to do operations that are inside of groups first. Well guess what… that means that when we're solving equations and undoing operations, we need to wait to do groups last of all. To see what I mean, let's solve this equation, which looks very similar to the first one we solved. The only difference is that a set of parentheses has been used to group this x plus 2 together. And even though that might not seem like much of a change, it makes a big difference for our answer. That's because in the original equation, this first 2 is only being multiplied by the x. But in the new equation, it's being multiplied by the entire quantity, or group, x plus 2. And that's going to change how we solve it. We're still going to follow our order of operations rules in reverse, but now that the x plus 2 is inside parentheses, which means that it's part of a group, we're going to undo that operation last. Since we're supposed to do operations in groups first, it means that we're going to undo operations in groups last. So in this problem, we should start by undoing the multiplication that's implied between the 2 and the group x plus 2. To do that, we divide both sides of the equation by 2. On the first side, the 2 on the top and the 2 on the bottom cancel, leaving the group x plus 2 on that side. And on the other side, we have 8 divided by 2 which is 4. That looks simpler already, and we can make it even simpler than that because now that there's nothing else on that side of the equal sign with the group x plus 2, we really don't even need the parentheses anymore. Next, we just need to subtract 2 from both sides. On the first side, the plus 2 and the minus 2 cancel out leaving x all by itself. And on the other side, we have 4 minus 2 which is 2. So for this equation, x equals 2. And now you can see how grouping operations differently in our equation results in different answers. Let's try one more important example. Do you remember the second equation we solved? x over 2 minus 1 equals 4. In this equation, the 1 is being subtracted from the entire x over 2 term. But take a look at this slightly different equation. This looks a lot like the original equation, but now that the 1 is up on top of the fraction line, it's only being subtracted from the x and not the 2. The x minus 1 on top forms a group. Hold on… how can x minus 1 be a group? I don't see any parentheses or brackets around them. Ah… that's a good question. In Algebra, the fraction line is used as a way to automatically group things that are above it or things that are below it. For example, in this fancy algebraic expression, everything that's on top of the fraction line forms a group, and everything on the bottom of the line forms another group. Of course, we could put parentheses there if we wanted to make it really clear, but it's not required. Grouping above and below a fraction line is just implied in Algebra. Getting back to our new problem, now that we know that the x minus 1 on the top of the fraction line is an implied group, as we learned in our last example, we're going to wait and undo the operation inside that group last. So the first step is to undo the divided by 2 by multiplying both sides of the equation by 2. On the first side, the 2 on the top and the 2 on the bottom will cancel out, leaving just our implied group x minus 1 on that side. And on the other side, we have 4 times 2 which is 8. Next, we can undo the operation inside the group by adding 1 to both sides. On the first side, the minus 1 and the plus 1 cancel, leaving x all by itself. And on the other side, we have 8 plus 1 which is 9. So in this equation, x equals 9. Alright, as you can see, solving two-step equations is definitely more complicated than single-step equations because there are so many different combinations and different ways to group things. But if you just take things one step at a time, and remember to undo operations using the reverse order of operation rules, it will be much easier. Just pay close attention to how things are grouped in an equation and be on the lookout for those implied groups on the top and bottom of a fraction line. And, because there are so many variations of these two-step equations, it's really important to practice by trying to solve lots of different problems. As always, thanks for watching Math Antics and I'll see ya next time. Learn more at www.mathantics.com