 Let's do a couple more equations to solve them and you know, see, just work with this a little bit more because you're going to get a lot of these things. And this is, you know, one of the main things with mathematics, right, solving equations. So let's say we have something like this. Now there are two ways to solve this. In general, when you start out, most of the equations you get are going to be fairly easy. And all the fractions or common denominators are going to work out, right? That's how you get started in mathematics. So the first method is initially going to be easier to do. The second method is cross multiplication when you're solving these types of things. And that's going to come in handy a lot more. You can end up using a lot more when it comes to more complicated questions. I'm in general going to use cross multiplication. Multiply whatever fractions and solve for it. The other way to solve this is multiply this equation by the common denominator. That is multiply every single term in the equation by the common denominator. And what that does is it eliminates all the denominators. It eliminates the fractions, right? So for example, all of these terms, anything that's not a fraction is over one, right? So that's over one, that's over one, and that's over one. Now what's the common denominator between one, two, one, three, three, and one? Well, that's just going to be six. So what you're going to do is multiply this whole equation by six. Okay? So when you multiply this whole equation by six, six multiplies every term here. X over one times six is just going to be six X. Five over two times six, two is going to kill the six down to three, right? Reduce the six down to three. Let's just do this on the side here. So we're going to have five over two times six. Five over two times six. That's just six over one. Anything from the bottom can count on anything from the top. Two kills six down to three. So it's just going to be five times three, which is going to be 15. So this becomes 15 minus, that's just one, one does nothing to six. So two times six is going to be 12. Now remember, when you're doing this, when the two kills reduces the six down to three, this six is going to multiply the next term. It's not the three. Each multiplication is an individual operation that you're doing. Over here, two X over three, three reduces the six down to two. So it's only a two multiplying this. So it's going to be four X minus three reduces the six down to two. So it's only a two going to be multiplying this. Two minus six, because it's just one times six. What you're going to do now is, again, line up your equals sign, combine your light terms on either side of the equation. So this is going to add to this. This is just going to be six X minus negative 15 minus 12 is negative 27. Four X minus 18. Now bring all your X's to one side of the equation. Take your numbers to the other side. So bring all your light terms to one side, the X's anyway. So it's going to be minus four X. Grab that guy, bring it over. So it's going to be plus 27. So six X minus four X is going to be two X. Over here, you've got 27 minus eight. That's going to be 19. Divide by two. Divide by two. So X is equal to 19 over two. And that's your final answer. Now over here, we didn't have to write down any restrictions because we didn't have any variables in the denominator. They were just numbers. And with numbers, there is no restriction. The only time you have restrictions if you have your variables, X's or whatever it is in the denominator, then then it can vary variable. So you have to state your restriction on it. So for this equation, our solution is X is equal to 19 over two. Now let's do the same equation with cross multiplication. It might add a couple more steps, but you'll see how it works. It's just later on with larger problems. Cross multiplication is going to be super handy. Initially, if you're doing these solving these types of equations, this is in general the way that I've come across most students are taught in classrooms. And when it comes to harder problems, one way you can become harder is if the common denominator between these is not something simple, if it becomes something gigantic. So for example, if you had the denominator, something like, I've got a couple more denominators, if you had a 7 here and a 5 here, all of a sudden the common denominator becomes a little harder to find. So this becomes a lot more difficult doing it this way as compared to cross multiplication because the common denominator is not obvious what it is. Let's do this solve the same problem using cross multiplication. So we've got the same problem. Now instead of multiplying this whole thing by 6 every single term by 6, you know, getting rid of denominators, right? What we're going to do is use cross multiplication. That means combine everything here into one term, combine everything there into one term, and then cross multiply if you still have fractions. And we're definitely going to still have fractions, right? So line up your equal sign. The common denominator on this side of the equation is 2. So that's just over 1 over 1. You multiply this by 2, so that becomes 2x. That just remains as 5 minus 4. On this side of the equation, the common denominator is 3. So that becomes 2x minus 1 minus 3. Now combine your like terms, right? So you've got line up your equal sign again. 2x negative 5 minus 4 is negative 9. Over 2 is equal to 2x minus 4 over 3. And what you do now is we've got one fraction equal to another fraction. We're going to cross multiply this out. And 3 is going to come up here and multiply both those terms. So it's going to be 6x minus 27. On this side it's going to be 4x minus 8. And the way we solve for this now is we grab this guy, bring it over. Plus 27, grab that guy, bring it over. Minus 4x. So 6x minus 4x is going to be 2x. And that's going to be 27 minus 8 is going to be 19. And divide by 2, divide by 2. So x is equal to 19 over 2. And that's how you solve this equation using cross multiplication. And this is basically in general going to be the method that I use because the equations that I come up with, I don't sit there and try to make as simple as possible. I just write down numbers and start doing them, right? I don't in general work them out initially to see if the answer is going to come out of the next equation or not. So sometimes I'm going to make equations up where the common denominator between the whole equation is extremely difficult to find. But it's going to be huge, right? Because they might not have the common denominator until you get further and further. Or you might have to multiply and hold everything together to get the common denominator. So in general, I'm going to use cross multiplication because I find that it works much better for solving equations, especially in real life, because in real life numbers don't work out as nicely as they do in the classroom. Or the questions that they give you are on an exam, right? In general numbers, you get a lot of irrational numbers. I've talked about irrational numbers. There's way more irrational numbers than rational numbers. And that's how you solve this equation using cross multiplication.