Solution - Linear equations with three unknowns

System of Linear Equations entered :

   [1]    2x + y - 3z = -8
   [2]    x - y + 2z = 12
   [3]    x - 2y + z = 6

Solve by Substitution :

// Solve equation [3] for the variable  z 
 

  [3]    z = -x + 2y + 6

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

   [1]    2x + y - 3•(-x +2y+6) = -8
   [1]    5x - 5y = 10

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

   [2]    x - y + 2•(-x +2y+6) = 12
   [2]    -x + 3y = 0

// Solve equation [2] for the variable  x 
 

  [2]    x = 3y 

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

   [1]    5•(3y) - 5y = 10
   [1]    10y = 10

// Solve equation [1] for the variable  y 

   [1]    10y = 10 

   [1]    y = 1 

// By now we know this much :

    x = 3y
    y = 1
    z = -x+2y+6

// Use the  y  value to solve for  x 

    x = 3(1) = 3 

// Use the  x  and  y  values to solve for  z 
  z = -(3)+2(1)+6 = 5 

Solution :

 {x,y,z} = {3,1,5}