Chapter 4 - Section 4.2 - Solving Systems of Linear Equations in Three Variables - Exercise Set - Page 221: 43

$(1,-1,2,3)$

$x+y+z+w = 5$ Equation $(1)$ $2x+y+z+w = 6 $ Equation $(2)$ $x+y+z =2$ Equation $(3)$ $x+y = 0$ Equation $(4)$ Using Equation $(4) $ in Equation $(3)$ $x+y+z =2$ $0+z =2$ $z =2$ Using $x+y = 0$ and $z =2$ in Equation $(1)$ $x+y+z+w = 5$ $0+2+w = 5$ $w = 5-2$ $w = 3$ Substituting $z$ and $w$ values in Equation $(2)$ $2x+y+z+w = 6 $ $2x+y+2+3 = 6$ $2x+y +5 = 6$ $2x+y =1$ Equation $(5)$ Subtracting Equation $(4)$ from Equation $(5)$ $2x-x+y-y = 1-0$ $x=1$ Substituting $x$ value in Equation $(4)$ $x+y=0$ $1+y=0$ $y = -1$ Solution:$(1,-1,2,3)$