Chapter 1.3, Problem 4E

In Exercises 1-4, transforms the augmented matrix for the given system to reduced echelon form and, in the notation of theorem 3, determine $n,r,$ and the number, $n-r,$ of independent variable. If $n-r>0,$ then identify $n-r$ independent variables.

$x_1+2x_2+3x_3+2x_4=1$

$x_1+2x_2+3x_3+5x_4=2$

$2x_1+4x_2+6x_3+x_4=1$

$-x_1-2x_2-3x_3+7x_4=2$