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CCSS.Math: , , , ,

a factory has mashite has machines that produce toys which are then packed by the factories workers one day each machine produced 14 toys and each worker packed 2 toys so that a total of 40 toys remained unpacked additionally the number of workers that day was 8 less than 7 times the number of machines how many machines and workers were there I encourage you to pause the video this is this is a good little little problem over here alright so let's let's define some variables so let's say M is equal to the number of machines and let's say that W is equal to the number of workers those seem like reasonable variables so what is the first sentence tell us it tells us one day each machine produced 14 toys so if each machine produced 14 toys what is the total number of toys that are going to be produced well the total number of toys that are going to be produced is going to be the number per machine times the number of machines so this is 14 M toys produced so this is what is produced produced right over there and then how many toys are going to be packed well if each worker packed two toys they tell us their each worker packed two toys so the total number that's going to be packed is going to be two toys per worker times the number of workers so that right over there that's the number of toys packed and then they tell us the number of toys that remain that the total that remains unpacked so the total it remains on fact we know that that is 40 let me do that in a neutral color so 40 40 so that we could view the 40 as produced but not packed produced produced not packed that's the number the total that remain unpacked well how do we how do we release produced and packed to the produce that are not packed well if we take the total that word used we subtract out the number that were packed we're going to be left with the total that are unpacked so just like that we were able to set up a linear relationship between M and W well just one isn't enough to solve for M and W but we have another relationship they say additionally the number of workers that day so the number of workers that day I could say W I'll write it over here W the number of workers that day was 8 less than 7 times the number of machines or you could say it was equal to 7 times the number of machines minus 8 that would be 8 less than 7 times the number of machines 7 M minus 8 and now we have two equations with two unknowns if things work out well we might be able to actually solve for W and M so there is a bunch of ways to do it since this one since this equation already has W explicitly solved for we can do some substitution here we can take this w and substitute it in for this w or actually I should say we could take 7m minus 8 and substitute it in for this w since the M and W the pair that we want to find you to satisfy both equations and so we are going to get we're going to get 14 m minus minus 2 minus 2 times let me do that in a so minus 2 times instead of a W I can write the 7m minus 8 so 7m minus 8 and we get that equals 40 is equal to 40 so we get now it's just a little bit of algebra 14 m and then let's see I'll do everything in a neutral color now so negative 2 times 7 M is negative 14 M and then negative 2 times negative 8 is plus 16 and then that's going to be equal to 40 now 14 M minus 14 M that's just going to be 0 and we're left at 16 is equal to 40 well that's never going to be true 16 is never going to be equal to 40 doesn't matter what M and W are in fact m and W have been eliminated from this equation this is impossible this right over here is impossible for 16 to be equal to 40 and because of that there are no solutions to this there are no there's no M and W pair that matches the constraints they gave us so there is no solution no solution I'll put that in a little square there