
 a) Write the chemical equation :

    a MnO[2] + b KOH +  c O[2] -> d K[2]MnO[4] + e H[2]O

   
 b) Write the list of the chemical elements.
 
         Mn :  a           =  d                            
          O : 2a + b + 2c  = 4d +  e       
          K :  b           = 2d           
          H :  b           = 2e            

 c) Rewrite the list.

         Mn :  a -  d                 = 0                            
          O : 2a +  b + 2c - 4d -  e  = 0       
          K :  b - 2d                 = 0           
          H :  b - 2e                 = 0                    


 d) Write the list of the chemical elements in a matrix :

           a     b     c     d     e    
           --------------------------
     Mn :  a                -d        = 0                            
      O : 2a     b    2c   -4d    -e  = 0       
      K :        b         -2d        = 0           
      H :        b               -2e  = 0                    

         

 e) Complete the matrix :

           a     b     c     d     e    
           --------------------------
     Mn : 1/1   0/1   0/1  -1/1   0/1  0/1                           
      O : 2/1   1/1   2/1  -4/1  -1/1  0/1      
      K : 0/1   1/1   0/1  -2/1   0/1  0/1          
      H : 0/1   1/1   0/1   0/1  -2/1  0/1                   
   



  f) This system has four rows, and five values to find.
     ( a, b, c, d ,e).
     If the system is consistent, it will have many solutions.

     To find a solution, I must give a value, to a coefficient.
     I have chosen to write a = 1/1.


           1,1,  0,1,  0,1,  0,1,  0,1, 1,1,         (a = 1/1)   
           1,1,  0,1,  0,1, -1,1,  0,1, 0,1,                           
           2,1,  1,1,  2,1, -4,1, -1,1, 0,1,      
           0,1,  1,1,  0,1, -2,1,  0,1, 0,1,          
           0,1,  1,1,  0,1,  0,1, -2,1, 0,1,                   
           








