Video Transcript: Two-Step Equations

Intro to two-step equations 

Now we mixed up things a little bit more: on the left side of the scale, not only do we have  these identical unknown masses with mass X, these three blue things, we also have some of the 1kg masses over here, actually, we have two of them. Now, we are going to figure out what X is. But before we even do that, I want you to think about a mathematical equation that can  represent what is going on; that equates what we have on the left hand, with what we have on  the right side of the scale. I will give you a few seconds to think about it... So let's think about  what we have on the left side: we have 3 masses with mass X, so you can say we have 3x and  then we have 2 masses of 1 kilogram, so in total we have 2 kg. So + 2. So one way to think  about the total mass on the left-hand side is 3x + 2. Three masses with mass X, plus two  kilograms. That is what we have on the left-hand side. Now, let us think about what we have on the right-hand side. We can simply count them: [counts to 14] Fourteen blocks, each has a mass of 1 kg, so the total mass will be 14 kg. And we see that the scale is balanced, not tilting down  or upwards. So this mass over here must be equal to this total mass. The scale is balanced, so  we can write an 'equal'-sign. (let me do that in a white coulour, I do not like that brown) Now,  what I want you to think about, and you can think about it either through the symbols or  through the scales, is: how would you go about -- let us think about a few things: how would  you first go about at least getting rid of these little 1kg blocks? I will give you a second to think  about that... Well, the simplest thing is: you can take these 1kg blocks off of the left-hand side,  but remember, if you just took these blocks off of the left-hand side, and it was balanced  before, now the left-hand side will be lighter and it will move up. But we want to keep it  balanced so we can keep saying 'equal'. That this mass is equal to that mass. So, if we remove 2  block from the left-hand side, we need to remove 2 from the right-hand side. So, we can  remove two there, and then we can remove two over there. Mathematically, what we are doing is: we are subtracting 2 kilograms from each side. We are subtracting 2 from this side, So on the left-hand side we now have 3x + 2, minus 2 we are left with just 3x, and on the right-hand side  we had 14 and we took away 2 (let me write this:) we took away 2, so we are going to be left  with 12 blocks. And you see that there, the ones that I have not crossed out, there are 12 left,  and here you have 3 of those X-blocks. Since we removed the same amount from both sides,  our scale is still balanced. And our equation: 3x is now equal to 12. Now, this turns into a  problem very similar to what we saw in the last video, so now I ask you: what can we do to  isolate one x, to only have one 'X' on the left-hand side of the scale, while keeping the scale  balanced? The easiest way to think about it is: If I want one X on this left-hand side, that is a  third of the total X's here. So what if I were to multiply the left-hand side by one-third -- -- but if I want to keep the scale balanced, I have to multiply the right-hand side by one-third. If we can  do that mathematically, Over here I can multiply the left-hand side by 1/3, and if I want to keep  my scale balanced I also have to multiply the right-hand side by 1/3. Multiplying it physically  literally means: just keeping a third of what we had originally We would get rid of two of these.  If we want to keep a third of what we had here originally, -- there are 12 blocks left over after  removing those first two -- so, 1/3 of 12: we are only going to have four of these little 1kg boxes  left. Let me remove all but four. (so, remove those, and those...) And I have left [counts them] 4 here. And so, what you are left with, the only thing you have left, is this 'X' - I will shade it in to  show this is the one we actually have left - and then we have these 1 kilogram boxes. You see it 

mathematically over here: 1/3 * 3x -- or you could have said 3x divided by 3 -- either way, that  gives us -- these threes cancel out, so that would give you an 'X' and on the right-hand side: 12  * 1/3 - which is the same as 12/3, is equal to 4. And, since we did the same thing to both sides,  the scale is still balanced. So you see that the mass of this thing must be the same as the mass  of these 4 left-over blocks. x must be equal to 4 kilograms. 

Two-step equations intuition 

Let's try some slightly more complicated equations. Let's say we have 3 times x plus 5-- I want  to make sure I get all the colors nice-- is equal to 17. So what's different about this than what we saw in the last video is, all of a sudden, now we have this plus 5. If it was just 3x is equal to 17,  you could divide both sides by 3, and you'd get your answer. But now this 5 seems to mess  things up a little bit. Now, before we even solve it, let's think about what it's saying. Let's solve  it kind of in a tangible way, then we'll solve it using operations that hopefully will make sense  after that. So 3 times x literally means-- so let me write it over here. So we have 3 times x. So  we literally have an x plus an x plus an x. That right there is a 3x. And then that's plus 5, and I'm  actually going to write it out as five objects. So plus 1, 2, 3, 4, 5. This right here, this 3x plus 5, is  equal to 17. So let me write the equal sign. Now, let me draw 17 objects here. So 1, 2, 3, 4, 5, 6,  7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. Now, these two things are equal, so anything you do to this  side, you have to do to that side. If we were to get rid of one object here, you'd want to get rid  of one object there in order for the equality to still be true. Now, what can we do to both sides  of this equation so we can get it in the form that we're used to, where we only have a 3x on the  left-hand side, where we don't have this 5? Well, ideally, we would just get rid of these five  objects here. You would literally get rid of these five objects: 1, 2, 3, 4, 5. But, like I said, if the  original thing was equal to the original thing on the right, if we get rid of five objects from the  left-hand side, we have to get rid of five objects from the right-hand side. So we have to do it  here, too: 1, 2, 3, 4, 5. Now, what is a symbolic way of representing taking away five things?  Well, you're subtracting 5 from both sides of this equation. So that's what we're doing here  when we took away 5 from the left and from the right. So we're subtracting 5 from the left.  That's what we did here. And we're also subtracting 5 from the right. Do that right over there.  Now, what does the left-hand side of the equation now become? The left-hand side, you have 5 minus 5. These cancel out. You're just left with the 3x. It's a different shade of green. You are just left with the 3x. The 5 and the negative 5 canceled out. And you see that here. When you got rid of these five objects, we were just left with the 3x's. This right here is the 3x. And the whole  reason why we subtracted 5 is because we wanted this 5 to go away. Now, what does the right hand side of the equation look like? So it's 3x is going to be-- let me write the equality sign right  under it-- is equal to-- or you could either just do it mathematically. Say, OK, 17 minus 5. 17  minus 5 is 12. Or you could just count over here. I had 17 things. I took away 5. I have 12 left: 1,  2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. That's what's subtraction is. It's just taking away five things. So  now we have it in a pretty straightforward form. 3x is equal to 12. All we have to do is divide  both sides of this equation by 3. So we're just left with an x on the left-hand side. So we divide  by-- let me pick a nicer color than that. Let me do this pink color. So you divide the left-hand  side by 3, the right-hand side by 3. And remember what that's equivalent to. The left-hand side, none of this stuff exists anymore, so we should ignore it. None of this stuff exists anymore. In 

fact, let me clear it out, just so that we don't even have to look at it. We subtracted it, so let me  clear it. Let me clear it over here. Let me clear it over here. And so now we are dividing both  sides by 3. Divide the left-hand side by 3. It's 1, 2, 3. So three groups, each of them have an x in  it. If you divide this right-hand side by 3, you have 1, 2 and 3. So it's three groups of four. So  when you do it mathematically here, the 3's cancel out. 3 times something divided by 3 is just  the something. So you're left with x is equal to, and then 12 divided by 3 is 4. You get x is equal  to 4, and you get that exact same thing over here. When you divided 3x into groups of three,  each of the groups had an x in it. When you divided 12 into groups of three, each of the groups  have a 4 in it, so x must be equal to 4. x is equal to 4. Let's do another one, and this time I won't  draw it all out like this, but hopefully, you'll see that the same type of processes are involved.  Let's say I have-- let me scroll down a little bit. Let's say I have 7x. So 7x-- and I'll do a slightly  more complicated one this time. 7x minus 2 is equal to-- I'll make the numbers not work out  nice and clean-- is equal to negative 10. Now, this all of a sudden becomes a lot more-- you  know, we have a negative sign. We have a negative over here, but we're going to do the exact  same thing. The first thing we want to do if we want to get the left-hand side simplified to just  7x is we want to get rid of this negative 2. And what can we add or subtract to both sides of the  equation to get rid of this negative 2? Well, if we add 2 to the left-hand side, these two guys will cancel out. But remember, this is equal to that. If we want the equality to still hold, if we add 2  to the left-hand side, we also have to do it to the right-hand side. So what is the new left-hand  side going to be equal to? So we have 7x, negative 2 plus 2 is just 0. I could write plus 0, or I  could just write nothing there, and I'll just write nothing. So we get 7x is equal to-- now, what's  negative 10 plus 2? And this is a little bit of review of adding and subtracting negative numbers.  Remember. I'll draw the number line here for you. If I draw the number line-- so this is 9, this is  1. We could keep going in the positive direction. Negative 10 is out here. Negative 10, negative  9, negative 8, negative 7. There's a bunch of numbers here. You know, dot, dot, dot. I don't have space to draw them all, but we're starting at negative 10, and we're adding 2 to it, so we're  moving in the positive direction on the number lines. So we're going 1, 2. So it's negative 8.  Don't get confused. Don't say, OK, 10 plus 2 is 12, so negative 10 plus 2 is negative 12. No!  Negative 10 minus 2 would be negative 12 because you'd be going more negative. Here, we  have a negative number, but we're going to the right. We're going in the positive direction, so  this is negative 8. So we have 7x is equal to negative 8. So now you might be saying, well, how  do I do this type of a problem? You know, I have a negative number here. You do it the exact  same way. If we want to just have an x on the left-hand side, we have to divide the left-hand  side by 7, so that the 7x divided by 7, just the 7's cancel out, you're left with x. So let's do that. If  you divide by 7, those cancel out, but you can't just do it to the left-hand side. Anything you do  to the left, you have to do to the right in order for the equality to still hold true. So let's divide  the right by 7 as well. And we are left with just an x is equal to negative 8 divided by 7. We could  work it out. It'll be some type of a decimal, if you were to use a calculator, or you could just  leave it in fraction form. Negative 8 divided by 7 is negative 8/7. Negative 8/7, or if you want to  write it as a mixed number, x is equal to 7 goes into 8 one time and has a remainder of 1, so it's  negative 1 and 1/7. Either one would be acceptable. 

Worked example: two-step equations

We have the equation negative 16 is equal to x over 4, plus 2. And we need to solve for x. So we  really just need to isolate the x variable on one side of this equation, and the best way to do  that is first to isolate it-- isolate this whole x over 4 term from all of the other terms. So in order  to do that, let's get rid of this 2. And the best way to get rid of that 2 is to subtract it. But if we  want to subtract it from the right-hand side, we also have to subtract it from the left-hand side, because this is an equation. If this is equal to that, anything we do to that, we also have to do to this. So let's subtract 2 from both sides. So you subtract 2 from the right, subtract 2 from the  left, and we get, on the left-hand side, negative 16 minus 2 is negative 18. And then that is  equal to x over 4. And then we have positive 2 minus 2, which is just going to be 0, so we don't  even have to write that. I could write just a plus 0, but I think that's a little unnecessary. And so  we have negative 18 is equal to x over 4. And our whole goal here is to isolate the x, to solve for  the x. And the best way we can do that, if we have x over 4 here, if we multiply that by 4, we're  just going to have an x. So we can multiply that by 4, but once again, this is an equation.  Anything you do to the right-hand side, you have to do to the left-hand side, and vice versa. So  if we multiply the right-hand side by 4, we also have to multiply the left-hand side by 4. So we  get 4 times negative 18 is equal to x over 4, times 4. The x over 4 times 4, that cancels out. You  divide something by 4 and multiply by 4, you're just going to be left with an x. And on the other  side, 4 times negative 18. Let's see, that's 40. Well, let's just write it out. So 18 times 4. If we  were to multiply 18 times 4, 4 times 8 is 32. 4 times 1 is 4, plus 1 is 72. But this is negative 18  times 4, so it's negative 72. So x is equal to negative 72. And if we want to check it, we can just  substitute it back into that original equation. So let's do that. Let's substitute this into the  original equation. So the original equation was negative 16 is equal to-- instead of writing x, I'm  going to write negative 72-- is equal to negative 72 over 4 plus 2. Let's see if this is actually true.  So this right-hand side simplifies to negative 72 divided by 4. We already know that that is  negative 18. So this is equal to negative 18 plus 2. This is what the equation becomes. And then  the right-hand side, negative 18 plus 2, that's negative 16. So it all comes out true. This right hand side, when x is equal to negative 72, does indeed equal negative 16.