From (R_2): (y + z = 3 \Rightarrow y = 3 - z) From (R_1): (x + 2(3 - z) + z = 2 \Rightarrow x + 6 - 2z + z = 2 \Rightarrow x - z = -4 \Rightarrow x = z - 4)
[ \left[\beginarrayc 1 & 2 & 1 & 2 \ 2 & 5 & 3 & 7 \ 1 & 3 & 2 & 5 \endarray\right] ]
Solve the system using elimination and back substitution: [ \begincases x + 2y + z = 2 \ 2x + 5y + 3z = 7 \ x + 3y + 2z = 5 \endcases ]
The last row says (0=0) (consistent system). Pivot variables: (x, y). Free variable: (z).
From (R_2): (y + z = 3 \Rightarrow y = 3 - z) From (R_1): (x + 2(3 - z) + z = 2 \Rightarrow x + 6 - 2z + z = 2 \Rightarrow x - z = -4 \Rightarrow x = z - 4)
[ \left[\beginarrayc 1 & 2 & 1 & 2 \ 2 & 5 & 3 & 7 \ 1 & 3 & 2 & 5 \endarray\right] ]
Solve the system using elimination and back substitution: [ \begincases x + 2y + z = 2 \ 2x + 5y + 3z = 7 \ x + 3y + 2z = 5 \endcases ]
The last row says (0=0) (consistent system). Pivot variables: (x, y). Free variable: (z).