Math 343 Fall12 Day 2 Sneetches, Thneeds and Green Eggs and Ham: An Application of Linear Systems of Equations (Solutions) The Story: Three neighboring communities developed a bartering system involving their primary industries. One community raised Sneetches. The second community manufactured Thneeds. The third community produced green eggs and ham. The Sneetch ranchers kept ¼ of the Sneetches for themselves, gave half of the Sneetches to the Thneed manufactures and gave ¼ of the Sneetches to the purveyors of green food. The Thneed manufactures kept 2/3 of their Thneeds and gave each of the other two communities 1/6 of their Thneeds. The purveyors of green food divided their green eggs and ham evenly amongst the three communities. Let’s denote Sneetch-­‐ranching community by S, the Thneed-­‐making community by T, and the makers of green eggs and ham by G. Task 1: Fill out the following table illustrating the bartering system. In each row, list what fraction of the product goes to each community. S T G Sneetches 1/4 1/2 1/4 Thneeds 1/6 2/3 1/6 Green Eggs & Ham 1/3 1/3 1/3 Now, these communites want to switch to a currency system. The idea is to assign a monetary value to each community’s total production for the year. But this is to be done in such a way that it matches the bartering system so that if the purchasing patterns match the bartering ratios then no one accumulates capital or debt. •
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Let x be the total value of the Sneetches produced each year. Let y be the total value of the Thneeds produced each year. Let z be the total value of the Green Eggs & Ham produced each year. Task 2: Assume that goods are distributed as with the barter system (indicated in the table above). For each of the three communities, write an expression that gives the total value of the amount of goods they receive each year (including the goods they made themselves and kept). 1
1
1
Value of goods received by S is x + y + z 4
6
3
1
2
1
Value of goods received by T is x + y + z 2
3
3
1
1
1
Value of goods received by T is x + y + z 4
6
3
Task 3: Now remember that we want the system to be fair. So the value that each community receives should be equal to the value that they produce. Using your results from Task 2, write a system of equations that expresses this condition of fairness. Setting each expression above equal to the value assigned to the goods produced by the community we get the following system of equations. 1
1
1
x + y + z = x 4
6
3
1
2
1
x + y + z = y 2
3
3
1
1
1
x + y + z = z 4
6
3
Rearranging so that all the variables appear on the left hand side of the equations, we get: −3
1
1
x+ y+ z = 0
4
6
3
1
−1
1
x + y + z = 0 2
3
3
1
1
−2
x+ y+
z=0
4
6
3
Task 4: Solve the system of equations in 3 different ways: 1) elimination method, 2) augmented matrix, 3) matrix algebra (just kidding, you won’t be able to do it this way) Elimination: −3
1
1
x+ y+ z = 0
4
6
3
1
−1
1
x + y + z = 0 2
3
3
1
1
−2
x+ y+
z=0
4
6
3
Let’s do some multiplication to get rid of all the fractions first: −9x + 2y + 4z = 0
3x − 2y + 2z = 0 3x + 2y − 8z = 0
Let’s add the first equation to 3 times the second equation and 3 times the third equation: −9x + 2y + 4z = 0
− 4y + 10z = 0 8y − 20z = 0
Now let’s add 2 times the second equation to the third equation. And make it prettier by multiplying the second equation by 1/2: −9x + 2y + 4z = 0
− 2y + 5z = 0 0=0
Check it out, we really only have two equations now to go with our three unknowns. This means that the system is under-­‐determined and so if it isn’t inconsistent, it will have infinitely many solutions. Which makes sense because you should be able to scale the price values without making things unfair (like make all dollars in the world worth 10 dollars). We can solve the second equation to see that y = 52 z Then substituting this back into the first equation we get -­‐9x +5z + 4z = 0. And solving this we see that x = z. So we get a different solution for every value of z that we want to choose. We could say that z = 100 Seuss bucks. Then x = 100 Seuss bucks and y = 250 Seuss bucks. Augmented matrix solution method: [This is just an alternative way to do the book-­‐keeping needed for the elimination method, so I am not going to bother writing it out here.] Task 5: Describe all the fair ways that the values of x, y, and z could be assigned. Any system where the value assigned to the total Sneetches produced is equal to the value of the total Green Eggs and Ham and the value assigned to the Thneed produced is 2.5 times as much will be fair. Task 6: Verify that under one such fair system that if the purchasing pattern matches the previous bartering system, then no community will accumulate capital or dept. Let’s use the pricing scheme I described when I solved Task 4. Suppose the value, x, of the Sneetch herd is 100 Seuss bucks, the value, y, of the Thneed’s produced each year is 250 Seuss bucks, and the value, z, of the Green Eggs and Ham produced is 100 Seuss bucks. Now suppose that the purchasing pattern matches the original barter system. Community S (the Sneetch ranchers) are going to keep ¼ of the Sneetchs which will be worth 25 Seuss bucks. Then they will receive 1/6 of the Thneeds, which is worth 125/3 Seuss bucks. They will also receive 1/3 of the Green Eggs and Ham which is worth 100/3 100
Seuss bucks. Thus they will end up with 25 +125
3 + 3 = 25 + 75 = 100 Seuss bucks worth of goods. Since they produced 100 Seuss bucks worth of Sneetches, they come out even for the year! Community T (the Thneed manufacturers and destroyers of the environment) will keep 2/3 of their Thneeds, which will be worth 500/3 Seuss bucks. They will receive ½ of the Sneetch herd, which will be worth 50 Seuss bucks. Finally, they will receive 1/3 of the Green Eggs and Ham, which will be worth 100/3 Seuss bucks. Thus they will receive a total 150
100
750
of 500
3 + 3 + 3 = 3 = 250 Seuss bucks of goods. And since they produce 250 Seuss bucks worth of Thneeds each year, they will come out even for the year!! Community G (Green Eggs and Ham producers) will keep 1/3 of their Green Eggs and Ham, which will be worth 100/3 Seuss bucks. They will receive 1/6 of the Thneeds which will be worth 125/3 Seuss bucks. Finally, they will receive ¼ of the Sneetches, which will be worth 25 Seuss bucks. This means they will receive a total of 100 Sneetch bucks worth of goods, which is exactly the value of the Green Eggs and Ham they will produce each year. So they come out even!