Solution - Linear equations with three unknowns

System of Linear Equations entered :

   [1]    x + y + z = 25
   [2]    5x + 3y + 2z = 0
   [3]    y - z = 6

Solve by Substitution :

// Solve equation [3] for the variable  y 
 

  [3]    y = z + 6

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

   [1]    x + (z +6) + z = 25
   [1]    x + 2z = 19

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

   [2]    5x + 3•(z +6) + 2z = 0
   [2]    5x + 5z = -18

// Solve equation [1] for the variable  x 
 

  [1]    x = -2z + 19

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

   [2]    5•(-2z+19) + 5z = -18
   [2]     - 5z = -113

// Solve equation [2] for the variable  z 

   [2]    5z = 113 

   [2]    z = 113/5 

// By now we know this much :

    x = -2z+19
    y = z+6
    z = 113/5

// Use the  z  value to solve for  x 

    x = -2(113/5)+19 = -131/5 

// Use the  z  value to solve for  y 

    y = (113/5)+6 = 143/5 

Solution :

 {x,y,z} = {-131/5,143/5,113/5}