 So at this point, we know everything we need to know in order to be able to solve any linear equation at all. Still, there are some details we may need to iron out. So let's try and solve the equation 8x plus 3 equals 12 minus 5x. Now, remember, trying to solve this equation means we want to get it into the form x equals stuff that doesn't include x. And the problem we see here is that there are variables on both sides. So we want to get all of our variables onto one side of the equation. So again, it's useful to identify what type of thing we have. Over on the left-hand side, we have a sum. Over on the right-hand side, we have a difference. And in the order of operations, these are done at the same time. And so that means that when solving equations, we can take care of either one first, your choice. Well, almost. Since I'm the one narrating the video, I get to make a choice. And whatever choice that I make is the one you're going to see. But in general, you can make either choice and you'll get to the same answer. And so this time, I will choose to take care of the difference. So on the right-hand side, we're subtracting 5x. So we'll take care of the difference by adding 5x. So now I have a new equation, 13x plus 3 equals 12. And now on the left-hand side, we have a sum. So I'm going to take care of the sum plus 3 by subtracting 3 from both sides. And now I have an equation, 13x equals 9. And 13x is a product of 13 and x. So I can get rid of the product by dividing by one of the factors. But never dividing by the variable. So I have to divide by 13. And I get my solution x equals 913. And of course, you should check your solution. Go ahead. I'll wait. Have a more complicated equation. So over on the left-hand side, I have a difference. Over on the right-hand side, I have both a sum and a product. The expression is actually a sum. But since we have a product, the order of operations requires us to deal with the product first. So that means I'm going to expand this product 3 times x minus 4. Here's where thinking about subtraction is adding the additive inverse becomes useful. 3x minus 12 plus 7 is the same as 3x plus additive inverse 12 plus 7. And addition can be done in any order that we want. So we can simplify the right-hand side. We can add this additive inverse 12 plus 7 to get additive inverse of 5. So the right-hand side is 3x plus additive inverse of 5. The left-hand side is still 5x minus 9. Now I have an addition and a subtraction. Since I eventually want to get this equation into the form x equals stuff, what I need to do is I need to get rid of the x terms on one side. Well, notice over on the right-hand side, I'm adding 3x. So I can undo that by subtracting 3x. And so now I have 2x minus 9 equals additive inverse of 5. Now over on the left-hand side, I have a difference. So I can take care of the subtraction by adding 9. Now we have 2x equals 4. That's a product times 2. So we'll take care of that by dividing by 2. And we get our solution x equals 2. While we can apply the preceding method to any linear equations, sometimes it's convenient to perform some operations first. And this emerges from the following idea. We can do almost anything we want to both sides of the equation. Most of the time it's not worth it. So if I have this perfectly tame equation, 1 fifth x plus 7 equals 3x minus 8, I could if I wanted to add square root of 17 to both sides. I'm not sure if I'd want to do that, so let's not. Well, is there something that it might be useful to do to both sides? And one of the possibilities here is if I multiply both sides by 5, then over on the left-hand side, the distributive property is going to give me x plus 35. While on the right-hand side, the distributive property is going to give me 15x minus 40. The reason this is useful is that anything that solves the original equation will solve the new equation. But the new equation is easier to work with. I know at this point some of you are saying, why would we do that? I like fractions. In fact, I wish every problem we had had nothing but fractions in it. I would do problems with fractions morning, noon, and night if I could. And if you feel that way, that's actually a good thing. Fractions are unfortunately inevitable, and so the more comfortable you are working with fractions, the better off you're going to be. Still, it can be sometimes difficult to work with them, so here's a useful idea to remember. If we multiply all terms of an equation by a common denominator, we can eliminate all fractions. So let's take this equation, 1 third x minus 2 equals 4 minus x. So the only denominator inside is this 3. So if we multiply every term by 3, we'll be able to eliminate all of the fractions. So let's do that. And we get this nice equation, x minus 6 equals 12 minus 3x. So we'll take care of the subtraction by adding 3x. We have another subtraction, 4x minus 6, so we'll take care of that by adding 6. We have a product 4 times x equals 18, so we'll deal with that by dividing by 4. And again, at this point, we've actually solved the equation because we now have this in the form x equals stuff that doesn't include x. Still, it is considered good style to reduce fractions if you can. And so this x equals 18 fourths is equal to...