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Use’s Excel Array Formulas to Solve

Simultaneous Equations

Simultaneous Linear Equation

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Present Example

The example we will consider here is from resistor circuits and goes by the name Kirchhoff’s circuit laws. The variables are currents in different

“loops” of the circuit. The coefficients (numbers multiplying

the variables) are resistances. The constants are voltages.

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Example with numbers

5 = 3.2 JA – 1.0 JB - 2.2 JC

0 = -1.0 JA + 5.5 JB – 3.0 JC

0 = -2.2 JA – 3.0 JB + 6.9 JC

In the above we have three current variables: JA, JB and JC.

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Loop equations as matrix equation

5 = 3.2 JA – 1 JB - 2.2 JC 0 = -1 JA + 5.5 JB – 3 JC 0 = -2.2JA – 3 JB + 6.9JC

0

0

5

9.632.2

35.51

2.212.3

C

B

A

J

J

J

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Enter matrix in Excel, highlight a region the same size as the matrix.

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In the formula bar, enter =MINVERSE(range) where range is the set of cells corresponding to the matrix (e.g. B1:D3). Then hit Crtl+Shift+Enter

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Result of matrix inversion

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Prepare the “voltage vector”, then highlight a range the same size as the vector and enter =MMULT(range1,range2) where range1 is the inverse matrix and range2 is the voltage vector. Then Ctrl-Shift-Enter.

Voltage vector

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Results of Matrix Multiplication

The answer

The current vector is the answer JA=3.152 mA

JB=1.470 mA

JC=1.644 mA

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Use the matrix approach in Excel to find the solution for 10 = 7 JA – 3 JB - 4 JC

0 = -3 JA + 6 JB – 1 JC

0 = -4 JA – 1 JB + 10JC

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Atomic Isotope Problem

Atomic MassesIsotopes:The element indium exists naturally as two isotopes. 113In has a mass of 112.9043 amu, and 115In has a mass of 114.9041 amu. The average atomic mass of indium is 114.82 amu. Calculate the percent relative abundance of the two isotopes of indium.

http://eppe.tripod.com/stoictry.htmPHY 202 (Blum) 13

Use the matrix approach in Excel to find the solution for 112.9043 x + 114.9041 y =

114.82x + y = 1where x is the fraction of the first isotope and x is the fraction of the second isotope.

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