Let

x = the number of horses

y = the number of cows

A farmer has a total of 45 animals:

x + y = 45

The ratio of horses to cows is 7:5

x : y = 7 : 5

5x = 7y

Then he added 18 horses, the new ratio is 8:1

(x + 18) : y = 8 : 1

x + 18 = 8y

By solving the system of equations:

x + y = 45

5x = 7y

x + 18 = 8y

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The system has not solutions thus the problem has no solutions as well.