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

If we solve this system of equations:

x+y=45

5x=7y

Click here to solve the system of equations

We get number of horses equal to 105/4 and number of cows equal to 75/4. 

Both are fractions, which can not be a number of either horse or cows (integers are expected).

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It's very likely that ratio 7:5 is misprint. Let's ignore it, and solve the problem without this constraint.

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So the total number of animals is still 45:

  x+y=45

After the farmer added 18 horses, the new ratio is 8:1

  (x+18) : y = 8:1

or

  x+18=8y

Let's solve the system of 2 equation we get:

  x+y=45

  x+18=8y

Click here to solve the system of equations

We get x=38, y=7.

So the final answer is: 38 horses and 7 cows.