108 CHAPTER 7. SYSTEMS OF EQUATIONS

row added to it. Thus the first step in solving 7.4 would be to take (−2) times the first rowof the augmented matrix above and add it to the second row, 1 3 6 | 25

0 1 2 | 80 2 5 | 19

 .

Note how this corresponds to 7.5. Next take (−2) times the second row and add to thethird,  1 3 6 | 25

0 1 2 | 80 0 1 | 3

This augmented matrix corresponds to the system

x+3y+6z = 25y+2z = 8

z = 3

which is the same as 7.6. By back substitution you obtain the solution x = 1,y = 6, andz = 3.

In general a linear system is of the form

a11x1 + · · ·+a1nxn = b1...

am1x1 + · · ·+amnxn = bm

, (7.7)

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

 . (7.8)

Changes to the system of equations in 7.7 as a result of an elementary operations translateinto changes of the augmented matrix resulting from a row operation. Note that Theorem7.1.4 implies that the row operations deliver an augmented matrix for a system of equationswhich has the same solution set as the original system.

Definition 7.1.6 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.

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.