Solution - Linear equations with three unknowns

System of Linear Equations entered :

   [1]    3x + 2y + z = 4
   [2]    x - y + z = 22
   [3]    -2x + 2z = 5

Solve by Substitution :

// Solve equation [2] for the variable  z 
 

  [2]    z = -x + y + 22

// Plug this in for variable  z  in equation [1]

   [1]    3x + 2y + (-x +y +22) = 4
   [1]    2x + 3y = -18

// Plug this in for variable  z  in equation [3]

   [3]    -2x + 2•(-x +? +22) = 5
   [3]    -4x = -39

// Solve equation [3] for the variable  ? 
 

  [3]    2? = 4x - 39

  [3]    ? = 2x - 39/2

// Plug this in for variable  ?  in equation [1]

   [1]    2x + 3•(2x-39/2) = -18
   [1]    8x = 81/2
   [1]    16x = 81

// Solve equation [1] for the variable  x 

   [1]    16x = 81 

   [1]    x = 81/16 

// By now we know this much :

    x = 81/16
    ? = 2x-39/2
    z = -x+?+22

// Use the  x  value to solve for  ? 

    ? = 2(81/16)-39/2 = -75/8 

// Use the  x  and  ?  values to solve for  z 
  z = -(81/16)+(-75/8)+22 = 121/16 

Solution :

 {x,?,z} = {81/16,-75/8,121/16}