2.2. GAUSS ELIMINATION 43

5. Divide each nonzero row by the value of the leading entry. The result will be a matrixin row reduced echelon form.

Sometimes there is no solution to a system of equations. When this happens, the systemis said to be inconsistent.

Here is another example based on the use of row operations.

Example 2.2.6 Give the complete solution to the system of equations, 3x− y− 5z = 9,y−10z = 0, and −2x+ y =−6.

The augmented matrix of this system is 3 −1 −5 90 1 −10 0−2 1 0 −6

After doing row operations, to obtain row reduced echelon form, 1 0 −5 3

0 1 −10 00 0 0 0

 .

The equations corresponding to this reduced echelon form are y = 10z and x = 3+ 5z.Apparently z can equal any number. Lets call this number t. 1Therefore, the solution set ofthis system is x = 3+5t,y = 10t, and z = t where t is completely arbitrary. The system hasan infinite set of solutions which are given in the above simple way. This is what it is allabout, finding the solutions to the system.

In summary,

Definition 2.2.7 A system of linear equations is a list of equations,

a11x1 +a12x2 + · · ·+a1nxn = b1

a21x1 +a22x2 + · · ·+a2nxn = b2...

am1x1 +am2x2 + · · ·+amnxn = bm

where ai j are numbers, and b j is a number. The above is a system of m equations in then variables, x1,x2 · · · ,xn. Nothing is said about the relative size of m and n. Written moresimply in terms of summation notation, the above can be written in the form

n

∑j=1

ai jx j = fi, i = 1,2,3, · · · ,m

It is desired to find (x1, · · · ,xn) solving each of the equations listed.

1In this context t is called a parameter.