a.$\quad$ x and y b.$\quad$ dependent c.$\quad$ $x=3+z,\ y=5-2z,\ z=$ any real number.
Work Step by Step
a. Leading variables correspond to leading entries in the reduced row-echelon form. x and y have a corresponding leading 1 in their correspoding columns, z does not. Answer: x and y. b. Since the last row represents an equation 0=0, which is always true, the system will have solutions. Since not all variables are leading, it will be dependent. Answer: dependent. c. Taking z to be any real number (since it has no corresponding leading entry), we back-substitute to find x and y. $x-z=3\quad\Rightarrow\quad x=3+z$ $y+2z=5\quad\Rightarrow\quad y=5-2z$ Answer: $x=3+z,\ y=5-2z,\ z=$ any real number.