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### Course: Operations and Algebraic Thinking 227-228>Unit 4

Lesson 5: Systems of equations word problems

# 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.
• Really useful when you give example questions that have no solution. Makes it really fun to try and work them out for ourselves, when you're trying to understand the concept but the question doesn't even have a solution. Thanks.
• Sometimes, we do encounter systems of equations that have no solution. It is good to be able to recognize the situation.
• Kind of wish I had known that "no solution" was an option before I paused the video and spent half an hour running in circles trying to solve it.
• so if this problem was in the Systems of equations word problems below, HOW CAN WE SOLVE THIS?
• As noted in the video, this system has no solution. A solution to a system of equations is the point(s) that the 2 lines have in common. These lines are parallel. They never touch each other, so they share no points in common. This is why the system has no solution.
• ok, I get how the linear equations work and how to get results out of systems of equations. but I dont know how to form/abstract an equation out of an real problem. how can I go on with the thought process?
• 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.
• 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