40 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS

Theorem 2.1.4 Given two equations involving the variables, (x1, · · · ,xn).

E1 = f1,E2 = f2 (2.2)

where E1 and E2 are expressions

E1 = a1x1 + · · ·+anxn

E2 = b1x1 + · · ·+bnxn

involving the variables and f1 and f2 are constants where the ai,bi, f1, f2 are in a fieldF. (In the above example there are only two variables, x and y and E1 = x + y whileE2 = 2x− y.) Then the system E1 = f1,E2 = f2 has the same solution set as

E1 = f1, E2 +aE1 = f2 +a f1. (2.3)

Also the system E1 = f1,E2 = f2 has the same solutions as the system, E2 = f2,E1 = f1. Thesystem E1 = f1,E2 = f2 has the same solution as the system E1 = f1,aE2 = a f2 provideda ̸= 0.

Proof: If (x1, · · · ,xn) solves E1 = f1,E2 = f2 then it solves the first equation in E1 =f1, E2 +aE1 = f2 +a f1. Also, it satisfies aE1 = a f1 and so, since it also solves E2 = f2 itmust solve E2 +aE1 = f2 +a f1. Therefore, if (x1, · · · ,xn) solves E1 = f1,E2 = f2 it mustalso solve E2 + aE1 = f2 + a f1. On the other hand, if it solves the system E1 = f1 andE2 + aE1 = f2 + a f1, then aE1 = a f1 and so you can subtract these equal quantities fromboth sides of E2+aE1 = f2+a f1 to obtain E2 = f2 showing that it satisfies E1 = f1,E2 = f2.

The second assertion of the theorem which says that the system E1 = f1,E2 = f2 has thesame solution as the system, E2 = f2,E1 = f1 is seen to be true because it involves nothingmore than listing the two equations in a different order. They are the same equations.

The third assertion of the theorem which says E1 = f1,E2 = f2 has the same solutionas the system E1 = f1,aE2 = a f2 provided a ̸= 0 is verified as follows: If (x1, · · · ,xn) is asolution of E1 = f1,E2 = f2, then it is a solution to E1 = f1,aE2 = a f2 because the secondsystem only involves multiplying the equation, E2 = f2 by a. If (x1, · · · ,xn) is a solutionof E1 = f1,aE2 = a f2, then upon multiplying aE2 = a f2 by the number 1/a, you find thatE2 = f2. ■

Stated simply, the above theorem shows that the elementary operations do not changethe solution set of a system of equations.

2.2 Gauss EliminationA less cumbersome way to represent a linear system is to write it as an augmented matrix.For example the suppose you want to find the solution for x,y,z in Z5 to the system

x+ 3̄y+ z = 0̄, 2̄x+ y+ 3̄z = 3̄,2̄y+ z = 4̄

To simplify, write the coefficients without the bar but do the arithmetic in Z5. 1 3 1 02 1 3 30 2 1 4

 .