Find the general solution of the linear system whose augmented matrix is

To start, I will divide the first row by 2:

Multiply the first row by -2 and add it to the third row:

Now divide the second row by -2:

Next, multiply the second row by 15 and add it the the third: row

The matrix is now in echelon form. To complete the process, I will transform it into reduced echelon form.

First, multiply the third row by -2/139:

Next, multiply the third row by 7/2 and add it to the second: row:

Now multiply the third row by -6 and add it the first:

Finally, multiply the second row by -5 and add it to the first: row:

Now the matrix is in reduced echelon form.

The first, third and fifth columns are the pivot columns. Therefore, the basic variables are x1, x3 and x5.

All other variables are free variables : x2 and x4 .

The general solution of this linear system is:

x1 = 288 / 139 - (2 * x2)

x2 is free

x3 = 34 / 139

x4 is free

x5 = 248/139

This underdetermined linear system is consistent and has an infinite number of solutions.