58 CHAPTER 4. SYSTEMS OF EQUATIONS

which is the same as 4.4. By back substitution you obtain the solution x = 1,y = 2, andz = 3.

In general a linear system is of the form

a11x1 + · · ·+a1nxn = b1...

am1x1 + · · ·+amnxn = bm

, (4.7)

where the xi are variables and the ai j and bi are constants. This system can be representedby the augmented matrix 

a11 · · · a1n | b1...

... |...

am1 · · · amn | bm

 . (4.8)

Changes to the system of equations in 4.7 as a result of an elementary operation translateinto changes of the augmented matrix resulting from a row operation. Note that Theorem4.1.5 implies that the row operations deliver an augmented matrix for a system of equationswhich has the same solution set as the original system.

Definition 4.1.7 The row operations consist of the following

1. Switch two rows.

2. Multiply a row by a nonzero number.

3. Replace a row by a multiple of another row added to it.

Example 4.1.8 In Exercise 4.1.6 some elementary operations were used to get the solutionto a system. here is the same example done with row operations. 1 −1 0 2

1 0 1 1−1 2 2 −1

 ,

 1 −1 0 20 1 1 −1−1 2 2 −1

 ,

 1 −1 0 20 1 1 −10 1 2 1

 ,

 1 −1 0 20 1 1 −10 0 1 2

 ,

 1 −1 0 20 1 0 −30 0 1 2

 ,

 1 0 0 −10 1 0 −30 0 1 2

So x =−1,y =−3,z = 2. In practice, you will make fewer mistakes if you don’t copy overthe matrix each time but simply erase and replace.

Gauss elimination is a systematic procedure to simplify an augmented matrix to a re-duced form. In the following definition, the term “leading entry” refers to the first nonzeroentry of a row when scanning the row from left to right.

Definition 4.1.9 An augmented matrix is in echelon form if

1. All nonzero rows are above any rows of zeros.