Solution - Linear equations with three unknowns

System of Linear Equations entered :

  [1]           2x+y-z=-3
  [2]           y+z=2
  [3]           4z-y+z=12

Equations Simplified or Rearranged :

   [1]    2x + y - z = -3
   [2]    y + z = 2
   [3]     - y + 5z = 12

Solve by Substitution :

// Solve equation [2] for the variable  z 
 

  [2]    z = -y + 2

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

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

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

   [3]     - y + 5•(-y +2) = 12
   [3]     - 6y = 2

// Solve equation [3] for the variable  y 

   [3]    6y = - 2 

   [3]    y = - 1/3 

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

   [1]    2x + 2•(-1/3) = -1
   [1]    2x = -1/3
   [1]    6x = -1

// Solve equation [1] for the variable  x 

   [1]    6x = - 1 

   [1]    x = - 1/6 

// By now we know this much :

    x = -1/6
    y = -1/3
    z = -y+2

// Use the  y  value to solve for  z 

    z = -(-1/3)+2 = 7/3 

Solution :

 {x,y,z} = {-1/6,-1/3,7/3}