Answer
a. $x=2,\quad y=1\quad z=3.$
b. $x=2-z,\ y=1-z,\ z=$ any real number.
c. $x=$none, $y=$none, $z=$ none.
Work Step by Step
a.
Each row of the matrix represents an equation.
$x=2,\quad y=1\quad z=3.$
b.
Each row of the matrix represents an equation.
The last row represents 0=0, so there will be solutions.
z does not have a corresponding leading 1, so we take it as a parameter (z= any real number),
we back-substitute to find x and y.
$x+z=2\quad\Rightarrow\quad x=2-z$
$y+z=1\quad\Rightarrow\quad y=1-z$
Answer: $x=2-z,\ y=1-z,\ z=$ any real number.
c.
Each row of the matrix represents an equation.
The last row represents 0=$3$,
an equation that no triplet x,y,z will satisfy,
so there will be no solutions.
Answer: $x=$none, $y=$none, $z=$ none.
