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# System of equations word problem: no solution

Systems of equations can be used to solve many real-world problems. In this video, we solve a problem about a toy factory. In this case, the problem has no viable solution, which means the information describes an impossible situation.

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• Ok. I understand that the scenario is imposible. According to previos vídeos there only could be 3 possible solutions graphing the system of equations. The only case with no solutions is represented by two parallel lines. Is this case graphed that way?
• It is exactly graphed that way; I used Desmos to input the equations and you can see that the lines are parallel: https://www.desmos.com/calculator/vkkpf2e7kw
• I did 14m+2w= -40 and got m= 24/28. Did i set this up incorrectly?
• Yes, sorry to say, you set it up incorrectly.
14m = total toys
2w = toys packed.
40 = toys remaining unpacked.
The key word is remaining. This is the result of a subtraction. As the workers pack the toys, they reduce the total toys. The 40 are the amount leftover where there weren't enough workers to pack them.
So, Sal's version: 14m - 2w = 40 is the equation you want.
OR, you can do: Total toys = Toys packed + Toys not yet packed. This creates the equation: 14m = 2w + 40.
Hope this helps.
• At , we could also convert the equation 14m - 2w = 40 in terms of w and express it in the slope intercept form.

We can observe that it is w = 7m - 20 while our other equation is 7m - 8. Hence, we can see that these are parallel lines with different y-intercepts and thus will never intersect, giving us a system with no solutions
• At we got two equations 14M(Toys) - 2W(Toys) = 40 (Toys) and 7M(workes)-W(Workers)=8(workers). How we can subtract workers from toys ?
• I went back and watched the video, so I do not understand what do you mean by having toys and workers in parentheses. When do you think he subtracts workers from toys?
The equations say 14M (machines) - 2W(workers) = 40 (toys produced, not packed) and W (workers) = 7M (machines) - 8. For the first equation, you need at least 3 machines, so 14(3) = 42 which would mean 4 machines created 42 toys, then with one worker 2(1) = 2 would mean that one worker packed 2 toys, so 40 toys left unpacked. 4 machines would produce 56, and 8 workers would pack 16 toys, so you are consistently talking about toys either created by machines or packed by workers. The second equation gives a comparison of number of workers and machines and has nothing to do with toys at all.
• Are systems of equations always linear? If so, why?
• A system of equations whose left-hand sides are linearly independent is always consistent.
• Is there shorter way or trick to analyze word problem?
• There are other ways of doing this, but I do not know if they are shorter or easier, we still have to interpret the words into math language:
the last sentence tells us that w=7m-8
From this, we can logically conclude that if m=1, we would have -1 workers, so this one does not make sense, so start at m=2 we would have 6 workers
We still need to interpret the first statement as 14m-2w = toys unpacked
Create a table of possible values
m 7m-8 w 14m-2w
2 7(2)-8 6 14(2) - 2(6) = 16
3 7(3)-8 13 14(3) - 2(13) = 42-26= 16
4 7(4)-8 20 14(4) - 2(20) = 56 - 40 = 16
5 7(5)-8 27 14(5) - 2(27) = 70 - 54 = 16

So we can come to the conclusion that it does not even matter how many machines and workers there are, with the equations we know, there will always be 16 unpackaged toys just as the video did when it ended up with 16=40 as an impossibility
• in the first equation 14m gives the no of toys produced by each machine .
then in the second 7m gives the no of machines
how can we equate these two?
• By defining variables, we are able to write equations.
If you watched the whole video, the 14m and the 7m were not in the same equation, but were related by a system of two equations
There was never an attempt to equate 14m and 7m, only relate these two through two equations.
• Is there a quicker way to find that there is no solution in this equation than doing the whole problem and finding out at the end? Thank you!