Answer
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.
