Solution - Linear equations with three unknowns

System of Linear Equations entered :

  [1]           2x-3y+z=-3
  [2]           x-5y+7z=-11
  [3]           -10+4y-6z=28

Equations Simplified or Rearranged :

   [1]    2x - 3y + z = -3
   [2]    x - 5y + 7z = -11
   [3]    4y - 6z = 38

Solve by Substitution :

// Solve equation [2] for the variable  x 
 

  [2]    x = 5y - 7z - 11

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

   [1]    2•(5y-7z-11) - 3y + z = -3
   [1]    7y - 13z = 19

// Solve equation [3] for the variable  y 
 

  [3]    4y = 6z + 38

  [3]    y = 3z/2 + 19/2

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

   [1]    7•(3z/2+19/2) - 13z = 19
   [1]     - 5z/2 = -95/2
   [1]     - 5z = -95

// Solve equation [1] for the variable  z 

   [1]    5z = 95 

   [1]    z = 19 

// By now we know this much :

    x = 5y-7z-11
    y = 3z/2+19/2
    z = 19

// Use the  z  value to solve for  y 

    y = (3/2)(19)+19/2 = 38 

// Use the  y  and  z  values to solve for  x 
  x = 5(38)-7(19)-11 = 46 

Solution :

 {x,y,z} = {46,38,19}