Construct an electrical circuit to find the values of Xa, Xb, and Xc in this system of equations:
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21Xa – 10Xb – 10Xc = 1
–10Xa + 22Xb – 10Xc = –2
–10Xa – 10Xb + 20Xc = 10
Your circuit should include only the following elements:
- one 1Ω resistor
- one 2Ω resistor
- three 10Ω resistors
- three ideal constant voltage sources
- three ideal ammeters
The circuit should be designed such that each ammeter displays one of the values, Xa, Xb and Xc.
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What is the numerical solution for the equations?
To solve the equations directly, start by solving the third equation for Xc and substituting it into the other two equations:
Xc = 1/2 Xa + 1/2 Xb + 1/2
21Xa – 10Xb – 5Xa – 5Xb – 5 = 1
-10Xa + 22Xb – 5Xa – 5Xb – 5 = -2
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16Xa – 15Xb = 6
-15Xa + 17Xb = 3
Solve for Xa by multiplying the first equation by 17 and the second equation by 15 and then adding them:
272Xa – 255Xb = 102
-225Xa + 255Xb = 45
47Xa = 147 → Xa = 147/47
Solve for Xb by multiplying the first equation by 15 and the second equation by 16 and then adding them:
240Xa – 225Xb = 90
-240Xa + 272Xb = 48
47Xb = 138 → Xb = 138/47
Finally, substitute those two results into the equation for Xc:
Xc = 147/94 + 138/94 + 47/94 = 332/94 = 166/47
