In class exercice 2/5/2002 Math 21b, Oliver KnillWe consider an electric circuitxyz90which contains a power supplyof 9 volts and is made of 5 resistors. In order to determine the voltage at each point, we need to know2 laws:• Ohms law U = RI relates the resistance R, the current I and the voltage difference U near aresistor.• Kirkhoffs law: tells that the sum of the currents passing to a point is zero.In our circuit all resistors have resistance 1. Consequently, by Ohm’s law the current through a resistoris the voltage difference. From Kirkhoff’s law we get the equations:x − 0 + x − y + x − z = 0y − x + y − z = 0z − x + z − 9 + z − y = 0The voltages x, y, z at the corresponding points of the circuit satisfy the linear equations:3x − y − z = 0−x + 2y − z = 0−x − y + 3z = 9Can you solve these equations?1In class exercice 2/5/2002 Math 21b, Oliver KnillFederal investigators in the collapse of Enron have found a network of firms or partners affiliatedwith Enron. Let us assume that the total loss is 9 billion dollars (in November, there were talks tosell Enron for 8 Billion to Dynegy).xyzEnronBank9 BiThe firms are tradingamong each other. If I is the cash flow between firms and assume that the sum of the money currentspassing through any firm is zero. If the firm network is as described in the picture, we obtain frommoney conservation the equations:x − 0 + x − y + x − z = 0y − x + y − z = 0z − x + z − 9 + z − y = 0The money losses x, y, z at the corresponding firms satisfy the linear equations:3x − y − z = 0−x + 2y − z = 0−x − y + 3z = 9Can you solve these equations to determine the loss of each of the
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