SOLUTION: The occupations of telephone operators, fishers, and sewing machine operators are among the ten with the largest Job decline from 2000-2012, according to the US department of labor

Question 564175: The occupations of telephone operators, fishers, and sewing machine operators are among the ten with the largest Job decline from 2000-2012, according to the US department of labor. The number of telephone operators will decline 8000 more than twice the number of fishers. The number of sewing machine operators will decline 1000 less than 10 times the number of telephone operators. If the total decline of these three jobs is predicted to be 137 thousand, find the predicted decline of each job.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
This is a strangely worded strange problem.
The strangest aspect of the problem is that the solution, as calculated, would not be a whole number, but that is the way this kind of prediction works. You have to take it about as seriously as the weather forecast. They may calculate that 1234.127 jobs will probably be lost in a certain occupation, and will report it as 1234.
The way I read it, the problem means to say the following:
The number of lost telephone operator jobs is expected to be twice the number of lost fisherman jobs plus 8000.
The number of lost sewing machine operator jobs is expected to be ten times the number of lost telephone operator jobs minus 1000.
Once we agree on what the problem meant, we can get to work to solve it. We can disagree on how to do that too, so I am providing two ways. You can figure out what solution your teacher expected.
THE GUESS AND CHECK WAY TO SOLVE IT (for elementary grades & others)
There are obviously less fishermen and less jobs were lost in that occupation.
{For my guess and checks, I make a table with a first column for the number of lost fisherman jobs and columns for the numbers of jobs lost in the other two occupations and a for the total of the 3 occupations. I write a guess for the first column, and calculate the corresponding numbers for the other columns. Each new guess fills one more line in the table. I cannot make a good table on this website, but I am including the explanations for all calculations and guessing decisions. The entire explanation below may look long and complicated, but the calculation work is not that complicated).
IF 1000 fisherman jobs were lost, the number of lost telephone operator jobs would be , and the number of lost sewing machine operator jobs would be . Then the total number of jobs lost among the three occupations would be 1,000+10,000+99,000=110,000.
Obviously the situation was worse than that. Let's try the next guess.
IF 2000 fisherman jobs were lost, the number of lost telephone operator jobs would be , and the number of lost sewing machine operator jobs would be . Then the total number of jobs lost among the three occupations would be 2,000+12,000+119,000=133,000.
Obviously the situation was worse than that, but we are getting close to the answer. Let's try the next guess.
IF 2200 fisherman jobs were lost, the number of lost telephone operator jobs would be , and the number of lost sewing machine operator jobs would be . Then the total number of jobs lost among the three occupations would be 2,200+12,400+123,000=137,600.
So the number of lost fisherman jobs must be between 2000 and 2200, closer to 2200. Let's try the next guess.
IF 2150 fisherman jobs were lost, the number of lost telephone operator jobs would be , and the number of lost sewing machine operator jobs would be . Then the total number of jobs lost among the three occupations would be 2,150+12,300+122,000=136,450.
So the number of lost fisherman jobs must be between 2150 and 2200. Let's try the next guess.
IF 2175 fisherman jobs were lost, the number of lost telephone operator jobs would be , and the number of lost sewing machine operator jobs would be . Then the total number of jobs lost among the three occupations would be 2,175+12,350+122,500=137,025.
So the number of lost fisherman jobs may be a little less than 2,175, but we are very, very close. Let's try the next guess.
IF 2173 fisherman jobs were lost, the number of lost telephone operator jobs would be , and the number of lost sewing machine operator jobs would be . Then the total number of jobs lost among the three occupations would be 2,173+12,346+122,460=136,979.
IF 2174 fisherman jobs were lost, the number of lost telephone operator jobs would be , and the number of lost sewing machine operator jobs would be . Then the total number of jobs lost among the three occupations would be 2,174+12,348+122,480=137,002.
So, 2173 lost fisherman jobs was too few, and 2174 was too many, but closer. That last try is as close as we will get, and that will be our solution, because we are not allowed to split fishermen.
USING A SYSTEM OF LINEAR EQUATIONS (for those studying systems of equations)
We can set up a system of linear equations
Let f be the number of lost fisherman jobs.
Let f be the number of lost fisherman jobs.
Let f be the number of lost fisherman jobs.
We write our equations:

Strangely enough, this system begs for substitution.
Substituting into we get
--> -->
Substituting and into we get
--> --> -->
We cannot split fishermen. Since , we take f=2174 as our approximation.
The rest is
--> -->
--> -->
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