Let x = the flat rate for the first 10 words.
Let y = the fixed charge per additional word.

Therefore to send an N word telegram costs, 
where N > 10, there are N-10 additional words.  
The cost would be:

Cost = (flat rate for first 10 words) + (No of additional words)×(fixed charge for each additional word)

or

Cost = x + (N-10)y

>>...a telegram of 15 words costs $11.65...<<

Plugging that in:

11.65 = x + (15-10)y or

11.65 = x + 5y 

or

>>...a telegram of 19 words costs $14.57...<<

Plugging that in:

14.57 = x + (19-10)y or

14.57 = x + 9y

So we have the system:

11.65 = x + 5y
14.57 = x + 9y

or

x + 5y = 11.65 
x + 9y = 14.57

Can you solve that? 

Answer x = 8 and y = .73

or 

Flat rate for the first 10 words = $8

Fixed charge for each additional word = $0.73

Edwin