Solution - Linear equations with three unknowns

System of Linear Equations entered :

  [1]           x+y+22=4
  [2]           2x-y+3z=9
  [3]           3x-y-z=2

Equations Simplified or Rearranged :

   [1]    x + y = -18
   [2]    2x - y + 3z = 9
   [3]    3x - y - z = 2

Solve by Substitution :

// Solve equation [1] for the variable  y 
 

  [1]    y = -x - 18

// Plug this in for variable  y  in equation [2]

   [2]    2x - (-x -18) + 3z = 9
   [2]    3x + 3z = -9

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

   [3]    3x - (-x -18) - z = 2
   [3]    4x - z = -16

// Solve equation [3] for the variable  z 
 

  [3]    z = 4x + 16

// Plug this in for variable  z  in equation [2]

   [2]    3x + 3•(4x+16) = -9
   [2]    15x = -57

// Solve equation [2] for the variable  x 

   [2]    15x = - 57 

   [2]    x = - 19/5 

// By now we know this much :

    x = -19/5
    y = -x-18
    z = 4x+16

// Use the  x  value to solve for  y 

    y = -(-19/5)-18 = -71/5 

// Use the  x  value to solve for  z 

    z = 4(-19/5)+16 = 4/5 

Solution :

 {x,y,z} = {-19/5,-71/5,4/5}