Solution - Linear equations with three unknowns

System of Linear Equations entered :

   [1]    3x + y + 2z = 3
   [2]    2x - 3y - z = -3
   [3]    x + 2y = 0

Solve by Substitution :

// Solve equation [3] for the variable  x 
 

  [3]    x = -2y 

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

   [1]    3•(-2y) + y + 2z = 3
   [1]     - 5y + 2z = 3

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

   [2]    2•(-2y) - 3y - z = -3
   [2]     - 7y - z = -3

// Solve equation [2] for the variable  z 
 

  [2]    z = -7y + 3

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

   [1]     - 5y + 2•(-7y+3) = 3
   [1]     - 19y = -3

// Solve equation [1] for the variable  y 

   [1]    19y = 3 

   [1]    y = 3/19 

// By now we know this much :

    x = -2y
    y = 3/19
    z = -7y+3

// Use the  y  value to solve for  x 

    x = -2(3/19) = -6/19 

// Use the  y  value to solve for  z 

    z = -7(3/19)+3 = 36/19 

Solution :

 {x,y,z} = {-6/19,3/19,36/19}