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# Algebra/help needed!

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

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.

Algebra

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#### Frederick Koh

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