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Current time:0:00Total duration:7:40

Video transcript

in many other videos we've taken equations like this and tried to quote solve for X what we're going to do in this video is deepen our understanding a little bit about what's going on and really think about the notion of equivalence or equivalent statements so let me write that down equivalence now what do I mean by equivalence well I'll use this equation here to essentially keep rewriting it in equivalent ways and I'll talk a little bit more about what that means so one thing that I could do to essentially write the same equivalent statement is I could distribute this 3 onto the X and or onto the X plus 1 and then this part of it could be re-written as 3x plus 3 and then we have minus X is equal to 9 now what I will say or what it might be obvious to you on some level is this top equation and this second equation are equivalent what does that mean it means if it if one of them is true for given X the other one will be true for that same X as well and vice versa and we can write other equivalent statements for example if we were to combine X terms here if we were to take 3x and then minus X right over there I could rewrite that as 2x and then I have plus 3 is equal to 9 now all three of these statements are equivalent if there's an X where 2 times that X plus 3 is equal to 9 then it is also the case that 3 times that X plus 1 minus X is equal to 9 and vice-versa if there's some X that would make this top equation true then it's going to make this last equation true and we can do other equivalents preserving operations and you've seen them before you could subtract 3 from both sides in general if you are adding or subtracting the same value from both sides it is equivalent equivalence preserving if you are distributing a value like we didn't in that first step that is equivalence preserving if you combine like terms so to speak that is equivalence preserving and so here we'll do an equivalence preserving operation we'll subtract 3 from both sides and you would get 2x is equal to 6 and once again any X that satisfies this last equation will satisfy any of the other equations and vice versa anything that's any exit satisfies and if any of these other equations will satisfy this last one and so they are all equivalent to each other and then another equivalence preserving operation is to multiply or divide both sides by a non-zero constant and so here we could divide both sides by 2 2 is not in 0 and it's constant and if we did that we will get another equivalent statement that X is equal to 3 so any X that satisfies this and there's one x equals 3 would satisfy the other ones and any X that satisfies any of the other ones would satisfy this last one so these are all equivalent so one way to think about it adding the same number to both sides of an equation that is equivalent preserving multiply or dividing both sides by a nonzero constant value is equivalent preserving distributing like we did in this first step equivalence preserving combining like terms equivalence preserving now you're probably saying well what are some non equivalence preserving operations well imagine something like this let me just start with something very obvious if I said that X is equal to 2 a non equivalence preserving operation is if I were to add or subtract or multiply or divide only one side of this equation by a value let's say I only added one to the left side then I would have X plus 1 is equal to 2 and it is not the case that anything that satisfies this second equation satisfies the top equation or vice versa x equals 2 clearly satisfies the top equation but it doesn't satisfy the second one that's because we did a non equivalence preserving operation likewise if I only multiplied the right-hand side by 3 I would get X is equal to 6 well by only multiplying the right-hand side by a value it's not the case that anything that satisfies x equals 6 will satisfy x equals two that is somewhat obvious here now there's a little bit trickier city scenarios let's say we have the equation 5x is equal to 6x now one temptation is well I want to do the same thing to both sides I could just divide both sides by X what will happen in that scenario well if you divide both sides by X you could think that an equivalent statement is that 5 is equal to 6 and you know that there's no X for which 5 is equal to 6 you can't make 5 equals 6 or 6 equal 5 and so this would somehow make you imply that okay there's no X that can satisfy if you assume that these were equivalent statements you'd say there's nothing that could satisfy 5 equals 6 so maybe there's nothing that satisfies this top equation but this actually isn't an equivalence preserving operation because you're actually dealing with the scenario where X is equal to 0 and you're dividing by 0 and so you have to be very careful when you're dividing by a variable especially if the variable that makes that equation true happens to be 0 and so in order to be clear that you do you're preserving equivalence here the way that I would tackle it is I would subtract 5x from both sides and if you do that and that is an equivalence preserving operation you could subtract that expression from both sides or that term from both sides and then you would be left with 0 is equal to X and now 0 equals x and 5 x equals 6x these are equivalent statements they are equivalent equations anything that makes this one true is going to make that one true and anyone MIT that makes that one true is going to make this one true now one last thing you might have heard me say you can multiply or divide by a non-zero value and that's going to be equivalence preserving and hopefully you just got a sense of why dividing by 0 is not a good idea in fact dividing by 0 is always going to be a strange thing and it's undefined but also multiplying by 0 for if I had let's say actually let me start over here if I had 2x is equal to six and if I were to multiply both sides by zero I would get zero is equal to zero and zero equals to certain zero is true for any X zero is always going to be equal to zero but the problem is is that first statement isn't true for all X it's only true for x equals three so these two are not equivalent statements they have a different set of X's that will satisfy them and so you have to be very careful when you're dealing with things that are either are zero you can add or subtract zeros obviously that's not going to change things much but when you multiply both sides by zero you can start getting things that are not equivalent statements and when you multiply or divide by things that could be zero like variables that also is a dangerous game to play