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Current time:0:00Total duration:9:28

Ex 3: Distributive property to simplify

Video transcript

like the last video I want to start with to warm up problems and then we'll do an actual word problem you're going to see these are going to be a little bit more involved than the equations in the last video but we're still going to be doing the exact same operations or what we could consider legitimate operations to get our answer so here we have 3 times X minus 1 is equal to 2 times X plus 3 so let's see what we can do here so the first thing I'd like to do and there's no definite right way to do it there are several ways you could do these problems but I like to distribute out the 3 and the 2 so 3 times X minus 1 that's the same thing as 3 X minus 3 I just distributed the 3 is equal to distribute out the 2 2 times X plus 2 times 3 which is 6 now we want to get what I like to do is get all of my constant terms on the same side of the equation and all my variable terms on the same side of the equation so let's see if we can get rid of this 2x term on the right hand side so let's subtract 2x I'm going to a slightly different notation this time because you might see it done this way or you might find it easier to visualize it this way it doesn't matter it's the same thing we did in the last video but I want to subtract 2x from this side of the equation but if I subtract 2x from this side of the equation I also have to subtract 2x from that side of the equation so then when we subtract 2x from both sides of the equation what we get here we get 3x minus 2x that's just 1 X or X minus 3x minus 3 is equal to well 2x minus 2x is no X's or 0 or and then you just have a 6 so we get X minus 3 is equal to 6 that was by getting rid of the 2x from the right hand side subtracting it from both sides of this equation now we have this negative 3 on the left hand side I don't want it there I just want an X there so to get rid of that we can add 3 to both sides of this equation we can add 3 to both sides of this equation you could imagine this is being adding this equation to the equation 3 is equal to 3 3 is obviously equal to 3 negative 2x is obviously equal to negative 2x you could view it either way a but if you add three to both sides of this equation the left-hand side of the equation becomes just an X because these two guys cancel out X equal and then 6 plus 3 is 9 and we are done and we can even check our answer 3 times 9 minus 1 is what this is 3 times 8 this is 24 so that's what the left-hand side equals what is the right-hand side equal that is 2 times 9 plus 3 that's 2 times 12 which is also equals 24 so it all works out X is equal to 9 next problem Z over 16 is equal to 2 times 3z plus 1 over all of that over 9 so it's this hairy looking problem one thing I let's multiply let's multiply both sides of this equation by 9 so if you multiply both sides of this equation by 9 what do we get we get 9 over 16 Z is equal to this 9 and that 9 will cancel out 2 times 3 Z plus 1 now let's distribute this 2 so we get 9 over 16 Z is equal to 2 times 3 Z is 6 Z plus 2 now let's get all of our Z's on the same side of the equation so let's subtract 6z from both sides of the equation so let's subtract minus 6z there and that of course equals minus 6z there and what do we get what do we get on this side on the left-hand side we get 9 over 16 minus 6 what is 6 if I have 16 as a denominator 6 is equal to what over 16 let me think about it 60 Plus 36 it's equal to 96 over 16 so it's 9 minus 96 over 16 Z this is just 6 I just rewrote minus 6 here is equal to these two cancel out that's why I subtracted 6z and the place so it's going to equal to so what is this equal right over here let me do an orange 9 minus 96 9 minus 96 well to see the difference between 9 and 96 is going to be 87 and of course we're subtracting the larger from the smaller so it's going to be negative 87 over 16 Z is equal to 2 and we're almost there we just have to multiply both sides of this equation by the inverse of this coefficient so multiply both side of this equation by 16 over or negative 16 over 87 negative 16 over 87 these cancel out 87 87 16 16 the negative signs plus you're just left with Z is equal to 2 times this thing so 2 times negative 16 is negative 32 over 87 and we are done that's a not a not a pretty-looking answer but that's what we got and you could try it with multiple methods doing maybe you can multiply both sides equation by 16 first maybe you can distribute out the 2 ninths first all sorts of things you can do and you can also verify that this is indeed the answer let's do a word problem now that we're warmed up all right it says Manoj and Tamar are arguing about how a number trick they heard goes tomorrow tells Andrew to think of a number let's say that X is the number that Andrew thinks of X think of a number multiply it by 5 so this is what Tamar is saying Tamar is telling Andrew to think of a number multiply it by 5 so let me do that so multiply it by 5 so he multiplied it by 5 and subtract 3 from the result and subtract we do that in a different color do it in blue subtract 3 from the result so subtract 3 from the result then Manoj comes along so here is Manoj Manoj comes along and tells Andrew once again to think of a number so once again think of a number we'll call that X so they both are trying to think of a number add five so now he's saying add five to the number so you add five to the number and then multiply the result and then multiply the result by three so multiply the result by three Andrew says that whichever way he does the trick he gets the same answer what was Andrews number so regardless of whether he does five times the number and then subtracts three or whether he adds five and then multiplies the whole answer by three he gets the same number so that must mean that these two things are equal he gets the same answer that means these two are equal so let's solve this equation we get 5x minus 3 is equal to 3 times X plus 5 a good place for me I like to distribute out this 3 so that is equal to 3x plus 15 I always have to remember to multiply the 3 times all of the terms in parentheses is a 15 and of course 5x minus 3 is equal to that now I think you know how I like to operate I like to get all of my X coefficients on one side so let's get them all on the left-hand side which means let's get rid of them on the right-hand side so let's subtract 3x from both sides so minus 3x minus 3x and then what do we get our equation becomes 5x minus 3x is 2x you still have a minus 3 is equal to these cancel out equal to 15 equal to that yellow 15 right there now we want to get rid of this negative 3 on the left-hand side and the best way to do it is to add 3 to both sides of the equation so add 3 to both sides of this equation so what do you get you now get 2x and these cancel out is equal to 18 and then you divide both sides of this by 2 you divide both sides by 2 and you get X is equal to 18 over 2 or 9 so in either situation Andrew was thinking of the number 9 let's see if it works according to both Tamar and manoah's method if I were to take five times if this is nine you get 5 times 9 is 45 minus 3 so you get 42 that's when you do Tamar's method when you do a monologist method this is a 9 right there 9 plus 5 is 14 so this thing becomes 14 and becomes 3 times 14 3 times 14 is 30 plus 12 which is 42 so andrew is right when he randomly picked the number 9 and regardless of which method he uses he gets the same result he gets 42