Solution - Linear equations with three unknowns

System of Linear Equations entered :

   [1]    2x + y - 4z = -7
   [2]    -x - 2y + 4z = 3
   [3]    2x - y + 2z = -1

Solve by Substitution :

// Solve equation [1] for the variable  y 
 

  [1]    y = -2x + 4z - 7

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

   [2]    -x - 2•(-2x+4z-7) + 4z = 3
   [2]    3x - 4z = -11

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

   [3]    2x - (-2x+4z-7) + 2z = -1
   [3]    4x - 2z = -8

// Solve equation [3] for the variable  x 
 

  [3]    4x = 2z - 8

  [3]    x = z/2 - 2

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

   [2]    3•(z /2-2) - 4z = -11
   [2]     - 5z/2 = -5
   [2]     - 5z = -10

// Solve equation [2] for the variable  z 

   [2]    5z = 10 

   [2]    z = 2 

// By now we know this much :

    x = z/2-2
    y = -2x+4z-7
    z = 2

// Use the  z  value to solve for  x 

    x = (/2)(2)-2 = -1 

// Use the  x  and  z  values to solve for  y 
  y = -2(-1)+4(2)-7 = 3 

Solution :

 {x,y,z} = {-1,3,2}