Solution - Linear equations with three unknowns

System of Linear Equations entered :

   [1]    2x + y - 3z = 25
   [2]    x - 2y + 2z = -22
   [3]    x - 3y + z = -19

Solve by Substitution :

// Solve equation [3] for the variable  z 
 

  [3]    z = -x + 3y - 19

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

   [1]    2x + y - 3•(-x +3y-19) = 25
   [1]    5x - 8y = -32

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

   [2]    x - 2y + 2•(-x +3y-19) = -22
   [2]    -x + 4y = 16

// Solve equation [2] for the variable  x 
 

  [2]    x = 4y - 16

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

   [1]    5•(4y-16) - 8y = -32
   [1]    12y = 48

// Solve equation [1] for the variable  y 

   [1]    12y = 48 

   [1]    y = 4 

// By now we know this much :

    x = 4y-16
    y = 4
    z = -x+3y-19

// Use the  y  value to solve for  x 

    x = 4(4)-16 = 0 

// Use the  x  and  y  values to solve for  z 
  z = -(0)+3(4)-19 = -7 

Solution :

 {x,y,z} = {0,4,-7}