Compare your answers to a and b with the answers for

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[Compare your answers to (a) and (b) with the answers for (a) and (b) in question 2. Also, re-try (b) and (c) with the 20 in (i) changed to 30, and compare your answers with that for 3.(d).] 4. Solving, you should have x = ( a 1 a 2 )/( b 2 b 1 ), or equivalently, x = ( a 2 a 1 )/( b 1 b 2 ), with y = ( a 1 b 2 a 2 b 1 )/( b 2 b 1 ) = ( a 2 b 1 a 1 b 2 )/( b 1 b 2 ). [These solution values for x and y apply generally. As an exercise, you may therefore want to try substituting in the values from some of the cases in questions 1-3, to con- M ATH M ODULE S olutions to Exercises 3
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firm that you get the correct answers. Do not feel obliged to memorize these formu- las; it’s almost always easier to work out the questions “from scratch.” What the for- mulas show, however, is that the solutions to every 2-equation linear system have exactly the same form. The formulas also show why inconsistent equations have no solution: the denom- inator for both x and y solutions is ( b 2 b 1 ), or ( b 1 b 2 ), where b 1 and b 2 are the slopes of the 2 equations in the system. When the 2 lines are parallel, b 1 = b 2 , and so the denominator is zero and the solution values for x and y are thus non-existent. 5. (a) P = $20/unit, Q = 4000 units. (b) P = $20/unit, Q = 400 units. (c) Rearranging, (i) P = 20 – 0.5 Q D and (ii) P = 30 + Q S . [This equation has a formal or mathematical solu- tion, with P = $23.33/unit and Q = – 6 2/3 units. If we are dealing with standard com- modities, however, then the meaning of negative quantities is unclear. Often, even if we do not state it explicitly, we assume that P ≥ $0 and that Q ≥ $0. In the present case, since there is zero quantity supplied at a price of $30/unit and no one is will- ing to pay more than $20/unit for positive amounts of the good, we can say that the non-negativity constraint ( Q ≥ $0) is binding, and that the quantity exchanged is zero. (This question was a “trick” question: its purpose is to remind us that even when our equations “work out” mathematically , we still need to ensure that they make econom- ic sense.)] 6. (a) Given P S = 20 + 0.1 Q S
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