Simultaneous Equations Can use matrix maths to solve linear simultaneous equations Arise from: –Electric Circuits Kirchoff’s laws –Fluid Dynamics

Systems of equations #1 Kirchoff’s Laws R0R0 R2R2 R1R1 ε0ε0 ε1ε1

Kirchoff’s Laws Kirchoff’s Node Law –The sum of currents at a node is equal to zero –Current does not build up at a node Kirchoff’s Loop Law –The sum of potentials around a loop is equal to zero

Kirchoff’s Laws Apply to circuit #1 R0R0 R2R2 R1R1 ε0ε0 ε1ε1 Loop1Loop2 I0 I1 I2

Kirchoff’s Laws Apply to circuit #2 Node Law –I0-I2-I1 = 0 Loop1 –I0R0-  0+I2R2=0 Loop2 –-I2R2-  1+I1R1=0

Kirchoff’s Laws Apply to circuit #3 Get rid of I2 & rewrite Loop 1 Loop 2

Kirchoff’s Laws & Simultaneous Equations Ohms law Also works with arrays of resistances & vectors of voltages & currents

Kirchoff’s Law Loop1 & Loop2 from before Write in matrix form

Kirchoff’s Laws Solution step Give values to resistance & voltages Form solution step

Systems of equations #2 Solution Step

Systems of equations #3

Systems of equations #4 From matrix toolbar