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CCSS.Math:

let's say we have the equation 7 times X 7 times X is equal to is equal to 14 now before even trying to solve this equation what I want to do is think a little bit about what this actually means 7x equals 14 this is the exact same thing as saying 7 times X let me write it this way 7 7 x times X to the X and orange again 7 times X is equal to 14 is equal to 14 now you might be able to do this in your head you could literally go through the 7 times tables you say well 7 times 1 is equal to 7 so that won't work 7 times 2 is equal to 14 so 2 works here so you would immediately be able to solve it you would be able to just by trying different numbers out say hey that's going to be a 2 but what we're going to do in this video is to think about how to solve this systematically because what we're going to find is as these equations get more and more complicated you're not going to be able to just think about it and do it in your head so it's really important that 1 you understand how to manipulate these equations but even more important understand what they actually represent this literally just says 7 times X is equal to 14 in algebra we don't write the x there when you write it two numbers next to each other or a number next to a variable like this it just means that you are multiplying it's just a shorthand a shorthand notation and in general we don't use the multiplication sign because it's confusing because X is the most is the most common variable used in algebra and if I were to write if I were to write 7 times X is equal to 14 if I write my time sign or my X a little bit strange it might look like X X or x times so in general when you're when you're dealing with equations especially when one of the variables is an X you wouldn't use the traditional multiplication sign you you you might use something like this you might use dot to represent multiplication so you might have 7 times X is equal to 14 but this is still a little unusual if you have something multiplying by a variable we'll just write 7x that literally means seven times X now to understand how you have to or how you can manipulate this equation to solve it let's think let's visualize this so seven times X what is that that's the same thing so I'm just going to rewrite this equation but I'm going to rewrite it in visual form so seven times X so that literally means X added to itself seven times that's the definition of multiplication so it's literally X plus X plus X plus X plus X let's see that's five X's plus X plus X so that right there is literally seven X's this is seven X right there let me rewrite down this right here this right here is is seven X now this equation tells us that 7 X is equal to 14 so the saying that this is equal to 14 let me just draw 14 objects here so let's say I have 1 2 3 4 5 6 7 8 9 10 11 12 13 14 so literally we're saying 7x is equal to 14 things these are equivalent statements now the reason why I drew it out this way is so that you really understand what we're going to do when we divide both sides by 7 so let me erase this right here so the the standard step whenever oh I didn't want to do that let me do this let me draw that last circle so in general whenever you simplify an equation down to a coefficient is just the number multiplying the variable so some number multiplying the variable or we could call that a coefficient times a variable equal to something else what you want to do is just divide both sides by 7 in this case or divide both sides by the coefficient so if you divide both sides by 7 what do you get 7 times something divided by 7 is just going to be that original something 7s cancel out and 14 divided by 7 is 2 so your solution is going to be X is equal to 2 but just to make very tangible in your head what's going on here is when we're dividing both sides of the equation by seven we're literally dividing both sides by seven this is an equation it's saying that this is equal to that anything I do to the left hand side I have to do to the right if they start off being equal I can't just do an operation to one side and have it still be equal they were the same thing so if I divide the left-hand side by seven so let me divide into seven groups so there are seven X's here so that's one two three four five six seven so it's one two three four five six seven groups now if I divide that into seven groups I'll also want to divide I'll also want to divide the right hand side into seven groups one two three four five six seven so if this whole thing is equal to this whole thing then each of these little chunks that we broke into these chunks these seven chunks are going to be equivalent so this chunk this chunk you could say is equal to that chunk this chunk is equal to this chunk they're all equivalent chunks there are seven chunks here seven chunks here so each X must be equal to two of these objects so we get X is equal to in this case in this case we had the objects drawn out where there's two of them X is equal to two now let's just do a couple more examples here just so it really gets in your mind that we're dealing with an equation in any operation that you do to one side of the equation you should do to the other so let me scroll down a little bit so let's say I have let's say I have 3x is equal to 15 now once again you might be able to do this in your head you're saying this is saying three times some number is equal to 15 you could go through your three times tables and figure it out but if you just wanted to do this systematically and it is good to understand it systemically you say okay this thing on the left is equal to this thing on the right what do I have to do to this thing on the left to have just an X there we'll have just an X there I want to divide it by three and my whole motivation for doing that is I have three times something the / 3 the 3s will cancel out and I'm just going to be left with an X now 3x was equal to 15 if I'm dividing the left side by 3 in order for the Equality to still hold I also have to divide the right side by 3 and what does that give us well the left-hand side we're just going to be left with an X so it's just going to be an X and then the right-hand side what is 15 divided by 3 well it is just 5 now you could have also done this equation in a slightly different way although they are really equivalent if I start with 3x is equal to 15 you might say hey Sal instead of dividing by 3 I could also get rid of this 3 I could just be left with an X if I multiply both sides of this equation by 1/3 so if I multiply both sides of this equation by 1/3 that should also work you say look 1/3 of 3 is 1 right when you just multiply this part right here 1/3 times 3 that is just 1 1 X 1 X is equal to 15 times 1/3 is equal to 5 and 1 times X is the same thing as just X so this is the same thing as X is equal to 5 and these are actually equivalent ways of doing it if you divide both sides by 3 that is equivalent to multiplying both sides of the equation by 1/3 now let's do one more and I'm going to make it a little bit more complicated so let's say and I'm going to change the variable a little bit so let's say I have 2 - y 2y plus 2y plus 4 plus 4y is equal to is equal to 18 now all of a sudden it's a little harder to do in your head we're saying 2 times something plus 4 times that same something is going to be equal to 18 so here it's harder to think about what number that is you could try and I say well if Y was 1 it would be 2 times 1 plus 4 times 1 well that doesn't work let's think about how to do it system you could keep guessing and you might eventually get the answer but how do you do this systematically let's visualize it so if I have two y's what does that mean it literally means i have to i have two y's added to each other so it's literally y plus y and then to that I'm adding four wise to that I'm adding four wise which are literally four wise added to each other so it's y plus y plus y plus y and that has got to be equal to 18 so that is equal to that is equal to 18 now how many Y's do I have here on the left-hand side how many Y's do I have I have one two three four five six wise so you could simplify this as 6y is equal to 18 and if you think about it makes complete sense so this is the whole this thing right here the 2y plus the four y is 6 Y so 2y plus four Y is 6 Y which makes sense if I have two apples plus four apples I'm going to have six apples if f two y's plus four wise I'm going to have six wise now that's going to be equal to 18 that is going to be equal to 18 and now hopefully we understand how to do this if I have six times something is equal to 18 if I divide both sides of this equation by 6 I'll solve for the something so divide the left-hand side by 6 divide the left-hand side by 6 and divide the right-hand side by 6 and we are left with Y is equal to Y is equal to 3 and you could try it out that's what's cool about an equation you can always check to see if you got the right answer let's see if that works 2 times 3/2 times 3 plus 4 times 3 is equal to what 2 times 3 this right here is 6 and then 4 times 3 is 12 6 plus 12 is indeed equal to 18 so it works out