78 CHAPTER 4. SYSTEMS OF EQUATIONS
10 volts
12 volts3 Ω
2 Ω
7 Ω
1 Ω
4 Ω
5 Ω 3 Ω
4 Ω2 Ω
I1 I2
I3
Those jagged places denote resistors and the numbers next to them give their resis-tance in ohms, written as Ω. The breaks in the lines having one short line and onelong line denote a voltage source which causes the current to flow in the directionwhich goes from the longer of the two lines toward the shorter along the unbrokenpart of the circuit. The current in amps in the four circuits is denoted by I1, I2, I3 andit is understood that the motion is in the counter clockwise direction. If Ik ends upbeing negative, then it just means the current flows in the clockwise direction. ThenKirchhoff’s law states that
The sum of the resistance times the amps in the counter clockwise direction arounda loop equals the sum of the voltage sources in the same direction around the loop.Find I1, I2, I3.
40. Here are some chemical reactions. Balance them.
(a) KNO3 +H2CO3→ K2CO3 +HNO3
(b) Ba3N2 +H2O→ Ba(OH)2 +NH3
(c) CaCl2 +Na3PO4→Ca3 (PO4)2 +NaCl
41. In the section on dimensionless variables 69 it was observed that ρV 2AB has the unitsof force. Describe a systematic way to obtain such combinations of the variableswhich will yield something which has the units of force.
42. A system of equations for x,y,z,w has augmented matrix1 −1 5 −4 0−1 2 −8 8 0−1 1 −5 4 00 −1 3 −4 0
Such systems are called homogenous systems because the column on the right is allzeros. Find all solutions to the system of equations having this as augmented matrix.Using the row reduced echelon form, show that the solutions to the underlying system