 In this video I'm going to talk about solving linear equations with variables on both sides. Sometimes when you have equations and you're solving you're going to have variables on the left and on the right side, the equal side. The most important thing to know when you are solving these type of equations is that you just need to gather everything together. Gather all the variables together, gather all the numbers together, that's what you need to do. In this case I see a lot of different terms. Here's a k term, 3k, negative 14k, and there's another k term, 25, that's a constant. Two is a constant, negative 6k, that's a k term, and then negative 12, that's another constant term. What I want to do is I want to gather everything together. First thing is on the left side I'm going to gather everything together that is alike. Notice that there's two k terms here, I'm going to add those together. Combined like terms that's going to be negative 11k plus 25. There's no other constant terms on this side so 25 just stays 25. Then on the right side I notice a 2 and a negative 12, those are both constant terms. So it's going to be negative 6k and then add those together to get a negative 10. Positive 2, negative 12 makes a negative 10. Okay, so that's the first thing you do is on both sides, left and right, make sure you kind of gather everything together. Now you need to make a decision. I need to get variables on one side of my equation and I need to get numbers on the other side. Now it doesn't matter what side, left, right, it doesn't matter just as long as they're on different sides. You just need to make a decision. Do I want the variables on the left or the variables on the right? It doesn't really matter, you just need to make that decision at this point. A good rule of thumb for most students is to always get the variables on the left side. So for this one that's what I'm going to do. I'm just going to get the variables on the left. So what I want to do is I want to take this negative 6k and I want to move it over to this other side. To take this and move it over I'm going to, since it's a negative 6k, I want to then add 6k, add 6k. Then in turn we'll kind of move this over to the other side. So that goes to zero. I'm left with negative 10 on that one side. Negative 11 plus 6 is a negative 5k plus 25. And notice the 25 didn't change at all. Now from there what I'm going to do is I'm going to get the numbers on the right side. Since I decided the variable goes on the left, the numbers, the constants are then going to go on the right side. So that means that this 25 needs to go to the other side. So positive 25 I need to subtract it over to the other side. So that means negative 5k is equal to, those are both negative numbers, that's a negative 35. So this is what I mean by get all your variables on one side and your numbers on the other. Now I've got one more step. I have a negative 5k here. I need to divide by negative 5 to get k by itself. So k is equal to, the negatives have canceled so this is going to be a positive 7. So k is equal to 7 in this case. Okay now again with any linear equation you can always take your answer and plug it back into your original equation to see if you did that correctly. This one's a little bit tougher. I can't do this in my head since there's a lot of variables, a lot of plugging in. 7 plug it back in. So 3 times 7 is 21. 3 times 14 is going to be 7098 so that's a negative 98. And then plus 25 equals 7 goes into here so this is going to be 2 minus 6 times 7 is 42 so that's a negative 42 minus 12. Okay so we're going to see if both the left and the right sides are the same. Now this is, again this is just to make sure that I did this correctly. Alright so the left side of this 21 and 25 is going to make 46. Okay and then 46 minus 98 that is 44 and 8 would be 52. Negative 52 excuse me. And then, okay so that's the left side of it, the right side, negative 42, negative 12 is 2 minus 54. This is a lot easier so this is going to be a negative 52. So notice the left and the right sides are both negative 52 so actually this is the correct answer. k in this case is equal to 7. Okay so that's kind of the way of solving linear equations. The biggest thing that you need to know is you just need to make a decision. I need to get my variables on one side and my constants on the other. Variables on one side, numbers on the other. Once you do that then you can start the process of moving things across the equal sign combining like terms, things like that. Okay so that's one example. I'm going to go through a couple more examples of this. One example, 3 times the quantity W plus 7 minus 5W equals W plus 12. Okay so on this one one thing that's a little bit different is that right here on the left side we do have a little bit of distributive property that we need to handle first. Okay so I'm just going to go through solving this. I'm going to take the 3 times everything in here so this is going to be 3 times W which is 3W, 3 times 7 which is a positive 21 minus 5W equals W plus 12. Notice nothing on the right side really needs to be changed. There's this W and 12 they can't combine so nothing happens there on the right side. Okay on the left side I see a 3W and a negative 5W. Those are two like terms so I'm going to add those together to make a negative 2W plus 21 equals W plus 12. And again now we've kind of come to the crossroads. I want my numbers on what side and my variables on what side. Well in this case actually what I'm going to do is now you don't always have to do this. You can choose what side everything's going to go to but I'm going to look at my variables to decide this. So I have a negative 2W on this left side and I got a W on the right side. What I'm going to do is I'm going to move the W's so that I have a positive amount of W's. What does that mean? Well if I take this W and I subtract it over to the left that means I'm going to have a negative 3W in the end. That negative 3 that means I'm going to have to eventually divide by negative 3. I don't really want to do that. If I look ahead a little bit I can try to make this a little bit easier for myself. So instead what I'm going to do is I'm actually going to do the opposite of this. I'm actually going to take this negative 2W and I'm going to move it over to the right side. That means I have to add it over here so that makes it a positive 3W. So I'll show you here in a second. So plus 2W plus 2W21 is equal to 3W plus 12. So notice that my variable has a positive number in it. That's why I kind of explained what I did. Does it really matter? Do I have to do it like that every time? No, no you don't. But once you get better and better with solving linear equations, solving equations in general, you always want to try to find little things like that, little strategies to help you out so that you make the solving just a little bit easier. Okay, moving on. Now I get the numbers over. So I'm going to subtract 12 from each side. What is that? 9 equals 3W. I'm actually going to move this up over here. I'm running out of room here on the bottom. 9 equals 3W. That means I need to divide by 3 on both sides. So 3 is equal to, there we go. And again I can plug that back into the top to see if I did that correctly. Okay, so 3 plus 7. Yep, W is 3. So 3 plus 7 is 10. 10 times 3, that's 30. So I got 30 right here. Minus 5 times 3, that's 15. So it's 30 minus 15. 30 minus 15 is, what's 30 minus 15? 15. 30 minus 15 is 15. So I got 15 there on the left side. I'm going to take W and plug it in here. 3 plus 12 is 15. So 15 on the left. 15 on the right. 15 on the right. It does work. Okay, I didn't do that correctly. All right. One more example. One more example to help us with this. One half times the quantity 10A plus 12 equals A minus 6. There we go. Okay, a lot of students will have a little bit of anxiety with this one because they have one half out front. Don't worry about that. It's not difficult to distribute a one half. Okay, all you got to do is just say to yourself, take one half of 10, one half of 12. That of word in mathematics means multiply. So half of 10 is 5, in this case 5A. Half of 12 is 6. And so again, not too difficult to multiply times one half. Okay. And after distributing, there isn't anything really to collect. These don't combine. These don't combine. So now I got to decide, okay, which side do I want my variables on? Which side do I want my numbers on? Okay, so in this case, I'm actually going to want my variables to be on the left. I'm going to subtract this A over. That's going to make it 4A. Notice it's going to be a positive 4A. That's what I want. I want a positive 4A. Equals. Nope, nope, sorry. I forgot about the 6. Got to bring the 6 down. And negative 6. Make sure you don't make that mistake. Just because this guy disappears doesn't mean this sign disappears. That sign tells us that 6 is negative. So make sure you keep that there. All right. And then what I need to do is subtract 6 from both sides. So 4A is equal to negative 6 and negative 6, making a negative 12. And then I need to divide by 4. A in this case is equal to negative 3. All right. Now, did I do that correctly? Let's check. Take this negative 3, plug it back in here. 10 times a negative 3 is negative 30. Negative 30 plus 12 is a negative 18. Half of negative 18. Half of negative 18 is going to be a negative 9. Let's write that down. Negative 9. Does that equal the right side? All right. So negative 3 minus 6. That does make negative 9. So it does work. It does work. Okay. All right. And that is solving linear equations with variables on both sides. Again, the big thing you need to remember is that just make the decision. Which side do I want my variable on? That's the biggest decision you need to make. After that, then you get your variables on one side. You get your numbers on the other. And then you just continue solving from there. Make sure that you always, always, always take your answer and plug it back into the equation. And that's 100%. You know that you did that problem right. You found the right answer when you plugged that back in. All right. And that is solving linear equations with variables on both sides and a couple of examples of them.