Help needed with word-number problem

Hello,

My math teacher gave us a worksheet with six 4-letter words.

Book = 13

Full= 19

Sail= 20

Boil = 11

Silk = 18

Foil = 10

Each letter in the word has a value and they all should equal the number after the word.

Also, the B for instance in the word Book has to be the same value as the B in the word Boil, so on and so forth.

I cannot figure this out after hours of trying. How do I do this? Thank you for your time!

Student X

You can formulate a system of linear equations to solve this.

If we let the values of the individual letters be represented by the letter variable themselves, 

then 

B+ 2O + K = 13 

F+ U+ 2L = 19 

S + A + I + L = 20 

B + O + I + L = 11 

S + I + L + K= 18 

F + O + I + L =10

The resulting augmented matrix can be written as such: 

A   B   F   I   K   L   O  S   U 

0   1   0   0   1   0   2  0   0       13 

0   0   1   0   0   2   0  0   1       19 

1   0   0   1   0   1   0 1   0     20 

0   1   0   1   0   1   1 0 0      11 

0   0   0   1   1   1   0 1   0      18 

0   0   1   1   0 1   1 0  0       10

A first tier reduction of the matrix gives

A= 12 - 1/2 O - 1/2 S

B= 3 - 3/2 O + 1/2 S

F= 2 - 3/2 O + 1/2 S

I= -1/2 - 1/4 O - 1/4 S + 1/2 U

K= 10 - 1/2 O - 1/2 S

L= 17/2 + 3/4 O - 1/4 S - 1/2 U

O= O

S = S

U=  U

Assuming all letters are associated with integer values, then a possible choice would be to let O=2, S=4 and U=6.

Then  A=9, B= 2,  F=1, I=1, K=7 and L=6. 

Naturally there will be many other possible solution sets since this system involves 9 variables yet is only described by 6 equations.

Hope this helps. Peace.

Best Regards,

Mr Koh