Solving Equations With Combining Like Terms

So if I look at number one, if I look here, I do have some like terms here. I have a two X to the three X, so I am going to combine them. If I combine two X plus three X, I get 5 X and that is going to equal to 45. Now I'm left with a simple one step equation. This is multiplication. So I'm going to divide to get rid of that 5. Whatever I do to one side, I do to the other. 45 divided by 5 is going to give me 9. So X is going to equal 9. For number two, drawing that line down. This time, we have no other variables to combine with our negative 7 X so that guy just stays alone. And we're going to just break them right down. Then we have a negative 5 and a positive two that we can combine. So we're going to do negative 5 plus two, gives me a negative three. And that's going to be equal to 24. And now we have a simple two step equation. 

Remember, or opposite of the order of operations, we get rid of the minus three first. By doing a plus three, get rid of that, we do a one side. I do to the other. So I do 24 plus three is 27. And that is going to be equal to a negative 7 X now I need to get rid of my negative 7. I'm going to divide by negative 7. On both sides, then I have 27 divided by negative 7. We're going to leave that right as a fraction. Easy way of using that fraction bar, we can leave this as a mixed number, or we can change it up sorry, leave it as an improper fraction, or we can change it into a mixed number in 7 goes into 27, three times, and we have 6 left over, denominator stays the same, and that is a negative, because my signs were different. Number three, we are working with some decimals now. It is still going to be the same process. Draw that line down the equal sign. 

So we separate our two sides of our equation. Look for our like terms. We have a 9.8 K, we have a negative 10.1 K so my signs are different here, which means I'm going to have to subtract. So I'm going to come over here and I'm going to do negative 10.1. And I've got a positive 9.8. I know I need to subtract those. So I can't do this. So I have to do borrow. Makes that an 11. So 11 -8 is three. Bring my decimal point straight down. 9 -9 is zero. I'm left with a 0.3 K, looking at my two numbers, my negatives have more, so it's a negative 0.3 K I don't have to do anything. There's nothing to combine with that negative 4.3. So they just comes down there. And then that is all going to equal to 26.3. Now we're going to get rid of my negative 4.3 by doing a positive 4.3 to both sides. Add it up. It's 6, ten, end up with a 30.6. And then I've still got my negative 0.3 K on this side. 

I need to divide by negative 0.3 on both sides of my equal sign. So now I have 30.6 divided by negative .3. So I'm going to come over here. Remember we slide our decimal point, so we're dividing by a whole number. Bring it straight up. Three goes into three once. One times three is three. Subtract it at zero. Bring down a zero, three can't go into zero. So that's zero here. Bring down my 6, three goes into 6, twice, and I end up with nothing left, no other numbers to bring down. So my answer is going to be K equals a negative since my signs are different. 100 two. For number four, draw my line down my equal sign. Check for my like terms. I have a 7.9 in a negative point, a negative 2.3 Y so I've got 7.9. And a negative 2.3, so we subtract them. 9 minus three is 6, 7 minus two is 5. Decimal point comes straight down. So I've got a 5.6 Y because my 7 was larger than my 2.3. Now I have a negative 14.4 and a positive two. 

I used up all my space over here. Negative 14.4 plus two member that is a 2.0, not a .2 when we line up our decimal points. Signs are different. We subtract. So I end up with a negative 12.4 because 14.4 has more. So minus 12 .4 is going to equal to negative 1.2. Now we're going to start solving our two step equation. We're going to get rid of our negative 12.4 by adding 12.4. Adding 12.4 over here. Come over here. We've got 12.4. -1.2 because my sides are different. Subtract maybe 11.2. And then it's going to be a positive because the 12.4 has more. So I'm left with 5.6 Y equals 11.2. Dividing both sides by 5.6. 11.2 divided by 5.6 decimal point moves over, bring straight up, come down here, we know it's going to we're hoping it's going to be more than once, so let's try twice, 56 times two is going to be 112. So then my Y is going to equal to two. 

Now for everybody's favorites, the fractions. For number 5. We have some T's that we need to combine. We have a one 8th T and a three fourths T, both positive, so I can just add them. Remember I need to have common denominators when I'm adding fractions. So my comments nominator between 8 and four is going to be 8, so one 8th can stay the same. Three fourths, we're going to have to multiply top and bottom by two to get to 8 for the denominator and then three times two is 6. Add them up. I get 7 8s. So I now have 7 eighths T nothing to combine my negative three ways, and that's going to equal to four. Get rid of my three by adding three to both sides. Four plus three is 7. I'm left with 7 eighths T on this side. Got to get rid of my 7 8s T by dividing by 7 8 T remember for those of you who remember to skip this step, you could also just multiply by the reciprocal and skip that one little step. So divide by 7 8 to 5 by 7 8s over here. Left with T equals, and then we're going to do 7, we're going to flip that 7 8s over and multiply. 

Cross cancel between my 7s, they become ones, and then T is just going to let me left with 8 over one when I multiply straight across, so T is going to just simply be 8. For number 6, I'm going to show you two different ways. The first way I'm going to show you is just leaving the fractions as is and dealing with all the fractions as they are. So first thing I'm going to do is I see some mixed numbers here and I don't want to deal with mixed numbers because when I'm dividing and multiplying I can't, I have to become improper fractions, so I'm just going to start right off and make them improper fractions. So two times 6 plus 5 is going to be 17 over 6. And then I'm just going to write down the rest of the problem. Minus two thirds N plus two times two is four plus one is 5 over two minus four N now I can start. Combining my like terms. 

For my like terms, I have a negative two thirds N and a negative four N, my signs are the same there, so I'm going to add them. So I've got a negative four over one and a negative two thirds. My common denominator is going to be a three. So two thirds is going to stay two thirds. Four over one, multiply by three, negative and four times three is 12. Signs are the same, so I'm going to add them up. So 12 plus two is going to give me 14 over three, both negative, so my answer remains negative. So I've got 17 over 6, and you've got negative 14, thirds, N, and now I have a negative two, and a positive 5 halves. So in a negative two, in a positive 5 halves, I'm going to have to subtract these two because my signs are different. 

Find my common denominator between one and two. It's going to be two. So 5 halves gets to stay the same. It's going to multiply by two, make this four over two, subtract these. I have 5 minus four is going to be one. Half, the 5 halves is bigger than the two. So it's going to be a positive one half. And we're here. So we've combined our light terms. Now we just need to solve our equation. So first step is going to be subtracting one half. From both sides. Remember I need to have that common denominator. So I'm just going to switch this over here to make this both into 6s. So one half, I'm going to multiply by three, and bring it over here. So it's going to be three over 6, and I've got 17 over 6. Let them subtracting three over 6, so 17 minus three. Is going to be 14 over 6. And then I'm left with a negative 14 thirds. And on this side. 

To divide by negative 14 thirds to make it go away. Divide by negative 14 thirds. And I'm going to come down here, and I'm going to do 14. Over 6 is going to be divided by 14 thirds. So it's 14 over 6 times three over 14. Our 14s cancel out. Three and 6, 6 becomes a two, three becomes a one. Multiply straight across. I have a one half sines were different. It's a positive and a negative, so my answer is negative, and I'm left with N is equal to one half. For those of you who do not want to deal with the fractions as much, you want to get rid of them. I'm going to show you another way to get rid of them on the back side of that same problem. So here I've written down this same problem. And now we're going to talk about that hint that I give you on the front side of the paper. It tells you that you can find a multiple of all of your denominators and multiply the entire equation by that number so that it gets rid of your fractions all across the board. So if I look at my denominators, I have 6, three and two. 

Well, I know that the number that goes into all three of those numbers is 6. So I'm going to multiply this whole entire equation times 6. It's easiest. Again, if I deal with my, if I have my mixed numbers here because I'm multiplying. So I'm going to rewrite this using my mix numbers. I mean, my improper fraction is not my mixed numbers. Okay. And then we're just going to multiply everything by 6. So if I come over here, I have to do 16 6 times 17 6. So 17 over 6 times 6 over one, notice those 6s cancel right out and become ones. I'm just left with 17 over one or 17. Boom, first fraction is gone. Now I've got to do 6 times negative two. That's easy. That's a negative 12. Now I have to do 6 times negative two thirds, so I've got two thirds, times 6. My three cancels out. Three goes into 6 twice. Multiply straight across. Left with four over one, which is just a four. It's still a negative. 

So it's a negative four N now I do my 6 times my 5 halves. Two of those into 6, three times, 5 times three is 15 over one, which becomes just plain old, positive 15. And then our last term, 6 times negative four. This is going to be negative 24 F now I have a equation with absolutely no fractions in it and I can just combine my like terms with just integers. So then I'm going to look at my variables. I have a negative four N and a negative 24 N, which is going to give me a negative 28 N then I have a negative 12 in a positive 15. Signs are different. So it's going to be a positive three because 15 is the larger. It has more. And that's still going to equal to 17. Subtract three to get rid of it. 

I'm left with 17 minus three is 14. And then you've got negative 28 N divide both sides by negative 28. And I'm left with N is equal to negative 14 over 28. That can be reduced by 14. So 14 divided by 14 is one, 28 divided by 14 is excuse me. Two, and it's a negative. So it's a negative one half, just like we got on the other side. And that is going to do it for solving the equations with combining like terms, have a good one.