Chapter 9 - Section 9.1 - Solving Systems of Linear Equations in Three Variables and Problem Solving - Exercise Set - Page 641: 53

$[1, 1, 0, 2]$

$x+y -w=0$ $ y+2z+w=3$ $x -z =1$ $2x-y -w=-1$ $x-z=1$ $2x-y-w=-1$ $x-z+(2x-y-w)=1+(-1)$ $x-z+2x-y-w=0$ $3x-y-z-w=0$ $x+y-w=0$ $3x-y-z-w=0$ $3x-y-z-w=x+y-w$ $3x-y-z-w-x-y+w=x+y-w-x-y+w$ $2x-2y-z=0$ $y+2z+w=3$ $2x-y-w=-1$ $y+2z+w+2x-y-w=3-1$ $y-y+2z+w-w+2x=2$ $2z+2x=2$ $2x+2z=2$ $(2x+2z)/2=2/2$ $x+z=1$ $x+z-z=1-z$ $x=1-z$ $2x-2y-z=0$ $2(1-z)-2y-z=0$ $2-2z-2y-z=0$ $2-3z-2y=0$ $x-z=1$ $x-z+z=1+z$ $x=1+z$ $x=1-z$ $x=1+z$ $1-z=1+z$ $1-z-1+z=1+z-1+z$ $0 = 2z$ $0/2=2z/2$ $0=z$ $x=1+z$ $x=1+0$ $x=1$ $2x-2y-z=0$ $2*1-2y-0=0$ $2-2y=0$ $2-2y+2y=0+2y$ $2=2y$ $2/2=2y/2$ $1=y$ $y+2z+w=3$ $1+2*0+w=3$ $1+0+w=3$ $1+w=3$ $1+w-1=3-1$ $w=2$