Video Transcripts: One-Step Equations

One-step addition & subtraction equations 

Now that we're comfortable with the "why" of why we do something to both sides of an  equation, let's see if we can apply it to some equations to solve for an unknown variable. So  let's say that you have x plus seven is equal to ten, and I want to solve for x. All its saying is  something plus seven is equal to ten, and you might be able to figure that out in your head, but  if you want to do it a little bit more systematically, you're like well just all I want on the left hand side is an x. Well if all I want on the left hand side is an x I'd want to get rid of the seven. I want  to subtract seven from the left-hand side, but if I want to maintain an equality here, whatever I  do to the left-hand side I also have to do to the right-hand side going back to our scales that's  so that we can keep our scale balanced, so that we can say that the left is still equal to the right. And so what we're going to be left with is x and then the sevens cancel out is equal to ten minus seven is equal to three. So that unknown is three, and you can verify it, three plus seven is  indeed equal to ten. Let's try one more. Let's say we have a minus five is equal to negative two.  So this is a little bit more interesting since we have all of these negative numbers here, but we  can use the exact same logic. We just want an a over here on the left-hand side so we have to  get rid of this negative five somehow. Well the best way of getting rid of a negative 5 is to add  five to it. So I'll do that. So I will add five to the left-hand side. But if I want the left-hand side to  stay equal to the right-hand side, whatever I do to the left I have to do the right. So I'm going to have to add five on the right-hand side as well, and so on the left-hand side I'm left with a, and  then the negative five and the positive five cancel out and on the right hand side, and they're  going to stay equal because I did the same thing to both sides, we have negative two plus five  which is equal to three. So a is equal to three. Once again you can verify it. Three minus five is  indeed equal to negative two. 

One-step addition equation 

Solve for a and check your solution. And we have a plus 5 is equal to 54. Now, all this is saying is  that we have some numbers, some variable a. And if I add 5 to it, I will get 54. And you might be able to do this in your head. But we're going to do it a little bit more systematically. Because  that'll be helpful for you when we do more complicated problems. So in general, whenever you  have an equation like this, we want to have the variable. We want this a all by itself on one side  of the equation. We want to isolate it. It's already on the left-hand side, so let's try to get rid of  everything else on the left-hand side. Well, the only other thing on the left-hand side is this  positive 5. Well, the best way to get rid of a plus 5, or a positive 5, is to subtract 5. So let's  subtract 5. But remember, this says a plus 5 is equal to 54. If we want the equality to still hold,  anything we do to the left-hand side of this equation, we have to do to the right side of the  equation. So we also have to subtract 54 from the right. So we have a plus 5 minus 5. Well,  that's just going to be a plus 0. Because if you add 5 and you subtract 5, they cancel out. So a  plus 0 is just a. And then 54 minus 5, that is 49. And we're done. We have solved for a. A is equal  to 49. And now we can check it. And we can check it by just substituting 49 back for a in our  original equation. So instead of writing a plus 5 is equal to 54, let's see if 49 plus 5 is equal to 54.  So we're just substituting it back in. 49, 49 plus-- let me do that in that same shade of green. 49 plus 5 is equal to 54. We're trying to check this. 49 plus 5 is 54. And that, indeed, is equal to 54.  So it all checks out.

One-step addition & subtraction equations: fractions & decimals 

Let's give ourselves some practice solving equations. So let's say we had the equation 1/3 plus A is equal to 5/3. What is the A that makes this equation true? If I had 1/3 plus this A, what does A  need to be in order for 1/3 plus that to be equal to 5/3? So there's a bunch of different ways of  doing this, and this is one of the fun things about equations is there's no exactly one right way  to do it. But let's think about what at least I think might be the simplest way. And before I work  through anything, you should always try to pause the video, and do it on your own. So what I  like to think about is can I have just my A on one side of the equation? And since it's already on  the left-hand side, let's see if I can keep it on the left-hand side, but get rid of this 1/3 somehow.  Well the easiest was I can think of getting rid of this 1/3 is to subtract 1/3 from the left-hand side of the equation. Now I can't just do that from the left-hand side of the equation. If 1/3 plus A is  equal to 5/3, and if I just subtract 1/3 from the left-hand side, then they're not going to be equal  anymore. Then this thing is going to be 1/3 less, which this thing isn't going to change. So then  this thing on the left would become less than 5/3. So in order to hold the equality, whatever I do on the left-hand side I have to do on the right-hand side as well. So I have to subtract 1/3 from  both sides. And if I do that, then on the left-hand side, 1/3 minus 1/3, that's the whole reason  why I subtracted 1/3 was to get rid of the 1/3, and I am left with A is equal to 5/3 minus 1/3, 5/3  minus 1/3, minus 1/3, and what is that going to be equal to? I have five of something, in this  case I have 5/3, and I'm gonna subtract 1/3. So I'm gonna be left with 4/3. So I could write A is  equal to 4/3. And you could check to make sure that works. 1/3 plus 4/3 is indeed equal to 5/3.  Let's do another one of these. So let's say that we have the equation K minus eight is equal to  11.8. So once again I wanna solve for K. I wanna have just a K on the left-hand side. I don't want  this subtracting this eight right over here. So in order to get rid of this eight, let's add eight on  the left-hand side. And of course, if I do it on the left-hand side, I have to do it on the right-hand side as well. So we're gonna add eight to both sides. The left-hand side, you are substracting  eight and then you're adding eight. That's just going to cancel out, and you're just going to be  left with K. And on the right-hand side, 11.8 plus eight. Well, 11 plus eight is 19, so it's going to  be 19.8. And we're done, and once again, what's neat about equations, you can always check to see if you got the right answer. 19.8 minus eight is 11.8. Let's do another one, this is too much  fun. Alright, so let's say that I had 5/13 is equal to T minus 6/13. Alright, this is interesting 'cause  now I have my variable on the right-hand side. But let's just leave it there. Let's just see if we  can solve for T by getting rid of everything else on the right-hand side. And like we've done in  the past, if I'm subtracting 6/13, so why don't I just add it? Why don't I just add 6/13? I can't just  do that on the right-hand side. Then the two sides won't be equal anymore, so I gotta do it on  the left-hand side if I wanna hold the equality. So what happens? So what happens? On the left hand side I have, let me give myself a little bit more space, I have 5/13 plus 6/13, plus 6/13 are  equal to, are equal to... Well, I was subtracting 6/13, now I add 6/13. Those are just going to add  to zero. 6/13 minus 6/13 is just zero, so you're left with T. So T is equal to this. If I have 5/13 and I  add to that 6/13, well I'm gonna have 11/13. So this is going to be 11/13 is equal to T, or I could  write that the other way around. I could write T is equal to 11/13.