Solution - Linear equations with three unknowns

System of Linear Equations entered :

   [1]    3x + 2y - z = 8
   [2]    -3x + 4y + 5z = -14
   [3]    x - 3y + 4z = -14

Solve by Substitution :

// Solve equation [3] for the variable  x 
 

  [3]    x = 3y - 4z - 14

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

   [1]    3•(3y-4z-14) + 2y - z = 8
   [1]    11y - 13z = 50

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

   [2]    -3•(3y-4z-14) + 4y + 5z = -14
   [2]     - 5y + 17z = -56

// Solve equation [2] for the variable  z 
 

  [2]    17z = 5y - 56

  [2]    z = 5y/17 - 56/17

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

   [1]    11y - 13•(5y/17-56/17) = 50
   [1]    122y/17 = 122/17
   [1]    122y = 122

// Solve equation [1] for the variable  y 

   [1]    122y = 122 

   [1]    y = 1 

// By now we know this much :

    x = 3y-4z-14
    y = 1
    z = 5y/17-56/17

// Use the  y  value to solve for  z 

    z = (5/17)(1)-56/17 = -3 

// Use the  y  and  z  values to solve for  x 
  x = 3(1)-4(-3)-14 = 1 

Solution :

 {x,y,z} = {1,1,-3}