Two-step equations intuition

Let's try some slightly more complicated equations. Let's say we have 3 times x plus 5-- I want to make sure I get all the colors nice-- is equal to 17. So what's different about this than what we saw in the last video is, all of a sudden, now we have this plus 5. If it was just 3x is equal to 17, you could divide both sides by 3, and you'd get your answer. But now this 5 seems to mess things up a little bit. Now, before we even solve it, let's think about what it's saying. Let's solve it kind of in a tangible way, then we'll solve it using operations that hopefully will make sense after that. So 3 times x literally means-- so let me write it over here. So we have 3 times x. So we literally have an x plus an x plus an x. That right there is a 3x. And then that's plus 5, and I'm actually going to write it out as five objects. So plus 1, 2, 3, 4, 5. This right here, this 3x plus 5, is equal to 17. So let me write the equal sign. Now, let me draw 17 objects here. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. Now, these two things are equal, so anything you do to this side, you have to do to that side. If we were to get rid of one object here, you'd want to get rid of one object there in order for the equality to still be true. Now, what can we do to both sides of this equation so we can get it in the form that we're used to, where we only have a 3x on the left-hand side, where we don't have this 5? Well, ideally, we would just get rid of these five objects here. You would literally get rid of these five objects: 1, 2, 3, 4, 5. But, like I said, if the original thing was equal to the original thing on the right, if we get rid of five objects from the left-hand side, we have to get rid of five objects from the right-hand side. So we have to do it here, too: 1, 2, 3, 4, 5. Now, what is a symbolic way of representing taking away five things? Well, you're subtracting 5 from both sides of this equation. So that's what we're doing here when we took away 5 from the left and from the right. So we're subtracting 5 from the left. That's what we did here. And we're also subtracting 5 from the right. Do that right over there. Now, what does the left-hand side of the equation now become? The left-hand side, you have 5 minus 5. These cancel out. You're just left with the 3x. It's a different shade of green. You are just left with the 3x. The 5 and the negative 5 canceled out. And you see that here. When you got rid of these five objects, we were just left with the 3x's. This right here is the 3x. And the whole reason why we subtracted 5 is because we wanted this 5 to go away. Now, what does the right-hand side of the equation look like? So it's 3x is going to be-- let me write the equality sign right under it-- is equal to-- or you could either just do it mathematically. Say, OK, 17 minus 5. 17 minus 5 is 12. Or you could just count over here. I had 17 things. I took away 5. I have 12 left: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. That's what's subtraction is. It's just taking away five things. So now we have it in a pretty straightforward form. 3x is equal to 12. All we have to do is divide both sides of this equation by 3. So we're just left with an x on the left-hand side. So we divide by-- let me pick a nicer color than that. Let me do this pink color. So you divide the left-hand side by 3, the right-hand side by 3. And remember what that's equivalent to. The left-hand side, none of this stuff exists anymore, so we should ignore it. None of this stuff exists anymore. In fact, let me clear it out, just so that we don't even have to look at it. We subtracted it, so let me clear it. Let me clear it over here. Let me clear it over here. And so now we are dividing both sides by 3. Divide the left-hand side by 3. It's 1, 2, 3. So three groups, each of them have an x in it. If you divide this right-hand side by 3, you have 1, 2 and 3. So it's three groups of four. So when you do it mathematically here, the 3's cancel out. 3 times something divided by 3 is just the something. So you're left with x is equal to, and then 12 divided by 3 is 4. You get x is equal to 4, and you get that exact same thing over here. When you divided 3x into groups of three, each of the groups had an x in it. When you divided 12 into groups of three, each of the groups have a 4 in it, so x must be equal to 4. x is equal to 4. Let's do another one, and this time I won't draw it all out like this, but hopefully, you'll see that the same type of processes are involved. Let's say I have-- let me scroll down a little bit. Let's say I have 7x. So 7x-- and I'll do a slightly more complicated one this time. 7x minus 2 is equal to-- I'll make the numbers not work out nice and clean-- is equal to negative 10. Now, this all of a sudden becomes a lot more-- you know, we have a negative sign. We have a negative over here, but we're going to do the exact same thing. The first thing we want to do if we want to get the left-hand side simplified to just 7x is we want to get rid of this negative 2. And what can we add or subtract to both sides of the equation to get rid of this negative 2? Well, if we add 2 to the left-hand side, these two guys will cancel out. But remember, this is equal to that. If we want the equality to still hold, if we add 2 to the left-hand side, we also have to do it to the right-hand side. So what is the new left-hand side going to be equal to? So we have 7x, negative 2 plus 2 is just 0. I could write plus 0, or I could just write nothing there, and I'll just write nothing. So we get 7x is equal to-- now, what's negative 10 plus 2? And this is a little bit of review of adding and subtracting negative numbers. Remember. I'll draw the number line here for you. If I draw the number line-- so this is 9, this is 1. We could keep going in the positive direction. Negative 10 is out here. Negative 10, negative 9, negative 8, negative 7. There's a bunch of numbers here. You know, dot, dot, dot. I don't have space to draw them all, but we're starting at negative 10, and we're adding 2 to it, so we're moving in the positive direction on the number lines. So we're going 1, 2. So it's negative 8. Don't get confused. Don't say, OK, 10 plus 2 is 12, so negative 10 plus 2 is negative 12. No! Negative 10 minus 2 would be negative 12 because you'd be going more negative. Here, we have a negative number, but we're going to the right. We're going in the positive direction, so this is negative 8. So we have 7x is equal to negative 8. So now you might be saying, well, how do I do this type of a problem? You know, I have a negative number here. You do it the exact same way. If we want to just have an x on the left-hand side, we have to divide the left-hand side by 7, so that the 7x divided by 7, just the 7's cancel out, you're left with x. So let's do that. If you divide by 7, those cancel out, but you can't just do it to the left-hand side. Anything you do to the left, you have to do to the right in order for the equality to still hold true. So let's divide the right by 7 as well. And we are left with just an x is equal to negative 8 divided by 7. We could work it out. It'll be some type of a decimal, if you were to use a calculator, or you could just leave it in fraction form. Negative 8 divided by 7 is negative 8/7. Negative 8/7, or if you want to write it as a mixed number, x is equal to 7 goes into 8 one time and has a remainder of 1, so it's negative 1 and 1/7. Either one would be acceptable.