Intro to equations with variables on both sides

let's try to solve a more involved equation so let's say that we have let's say that we have let me think I'm good say we have 2 X plus 3 2 X plus 3 is equal to is equal to 5 X is equal to 5 X minus 2 so this might look a little daunting at first we have X's on both sides equations we're adding and subtracting numbers how do you solve it now we'll do it a couple of different ways the the important thing to remember is we just want to isolate an X once you've isolated and actually have x equals something or x equals something you have you're done you've solved the equation you can actually go back and check whether that works so what we're gonna do is just do a bunch of operations on both sides of the equation to eventually isolate the X but while we do those I actually want to visualize what's occurring cuz I don't want you to just say oh what are the rules or the steps of solving equations and I forgot whether this is a lot or that isn't allowed if you visualize what's happening it'll actually be common sense what's allowed so let's visualize it so we have 2x right here on the left-hand side so that's literally that's X plus X and then you have plus 3 plus 3 I'll do it like this so that's equivalent to plus 1 plus 1 plus 1 that's the same thing as 3 I could have drawn 3 circles here as well but let me do the same color plus 3 and then that is equal to 5 X's I'll do that in blue that is equal to 5 X's so 1 2 3 4 5 and I want to make it clear you would never actually have to do it this way when you're solving the problem you would just have to do the algebraic steps but I'm doing this for you so you can actually visualize what this equation is saying the left-hand side is these two orange X's plus 3 the right-hand side is 5x minus 2 so minus 2 we could write as minus so let me do this in a different color I'll do it in pink so minus 2 I'll do as minus 1 and minus 1 now we want to isolate the X's on on the same side of the equation so how could we do that well there's two ways of doing it we could subtract these two exes from both sides of the equation and that would be pretty reasonable because then you'd have five X's minus the two x's you would have a positive number of X's on the right-hand side or you could actually subtract 5x from both sides and that's what's neat about algebra as long as you do legitimate operations you will eventually get the right answer so let's just start off subtracting 2x from both sides of the equation and what I mean that I mean we're gonna remove two X's from the left hand side and if we remove two hat exits from the left hand side we have to remove two x's from the right hand side just like that so what does that give us we're subtracting 2 X's 2 XS from the left and we're also going to subtract 2 X's from the right now what is what does our left hand side simplify to we have 2x + 3 minus 2x the two x's cancel out so you're just left with you're just left with the 3 and you see that over here we took two of these X's away we're just left with the plus 1 plus 1 plus 1 and then on the right hand side 5x minus 2x we do have it over here with five X's minus 2 X's you only have one two three X's left over three is equal to 3x and then you have your minus 2 there you have your minus 2 so normally if you were to do the problem you just have to write what we have here on the left hand side so what can we do next remember we want to isolate the X's but we have all of our X's on the right hand side right here if we could get rid of this negative two off of the right hand side then it the x's will be alone they'll be isolated so how can we get rid of this negative two or if we visualize it over here this negative one this negative 1 well we could add 2 to both sides of this equation think about what happens there so if we add two so I'm going to do it like this plus 1 plus 1 so you can literally see we're adding 2 and then we're gonna add 2 to the left hand side 1 plus 1 plus what happens well let me do it over here as well so we're gonna add 2 we're gonna add 2 so what happens to the left hand side 3 plus 2 is going to be equal to 5 and that is going to be equal to 3 X minus 2 plus 2 these guys cancel out and you're just left with 3x and we see it over here we the left hand side is one plus one plus one plus one plus one we have five ones or five and the right-hand side we have the three X's right over there and then we have the negative one negative one plus one plus one negative one these cancel out they get us to zero they cancel out so we're just left with five is equal to three X so we have one two three four five is equal to three X let me clear everything that we've removed so it looks a little bit so it looks a little bit cleaner let me clear that out and then let me clear these are all of the things that we've removed let me clear that out and then let me clear that out like that edit clear so now we are just left with one two three four five lectures let me move this over so edit cut edit paste so I could just move this over right over here we now have we don't have 1 2 3 4 5 these are the two that we added here is equal to three X's these guys cancelled out that's why we have nothing there now to solve this we just divide both sides of this equation by 3 and this is going to be a little hard to visualize over here but if we divide over here both sides by 3 what do we get we divide the left by 3 we divide the right by 3 the whole reason why we divided by 3 is because the X was being multiplied by 3 3 is the coefficient on the X fancy word it literally just means the number multiplying the variable the number we're solving the variable we're solving for so these threes cancel out the right-hand side of the equation is just X the left hand side is 5/3 so 5/3 we could say X is equal to 5/3 and this is different than everything we've seen so far I now have the X on the right hand side the value on the left hand side that's completely fine this is the exact same thing as saying 5/3 is equal to X is the same thing as saying X is equal to 5/3 completely equivalent completely equivalent we sometimes get more used to this one but this is completely the same thing now if we wanted to write this as a mixed number if we want to write this as a mixed number 3 goes into 5 one time with remainder 2 so it's going to be 1 & 2 Thirds so it's going to be one and two thirds so we could also write that X is equal to one and two thirds and I'll leave it up for you to actually substitute back into this original equation and see that it works out now to visualize it over here you know how did you get one in two thirds let's think about it instead of doing one I'm gonna do circles I am going to do circles actually even better I'm gonna do squares so I'm gonna have five squares on the left-hand side I'll do it in this same yellow color right here so I have one two three four five and that that is going to be equal to the three X's X plus X plus X now we're dividing both sides of the equation by three we're dividing both sides of the equation by three actually that's where we did up here we divide both sides by three so how do you do that right hand side is pretty straightforward you want to divide these three X's into three groups that's one two three groups one two three now how do you divide five into three and when they have to be thought through even groups and here tell them the answer tells us each group is going to be one in two thirds so one and two thirds this is going to be two thirds of this next one and then we're gonna have one in two thirds so this is one-third we're gonna need another another one so there's one in one-third we're gonna need one more of thirds so it's gonna be right here and then we're left with 2/3 and one so we've broken it up into three groups this right here let me make it clear let me make it clear this right here is one and two thirds one and two thirds and then this right here this is one-third that's another one-third so that's two thirds and then that's one right there so that's one and two thirds and then finally this is two thirds and this is one so this is 1 and 2/3 so when you divide both sides by 3 you get 1 and 2/3 each each section each each bucket is 1 and 2/3 on the left-hand side on the left-hand side or 5/3 and on the right-hand side we just have an X so it still works a little bit harder to visualize with fractions