>>...Isaac is nine years older than Ruth...<< That says:

, we start with Ruth's age, R, and then add 9

>>...Isaac is four times as old as Brian was
three years ago...<<  That says:

Three years ago Brian was , we start with Brian's age,
B, and subtract 3.

so Isaac is four times , so

, we multiply  by .

>>...Brian is eighteen years younger than Issac...<<

That says:

, we start with Isaac's age I and then subtract 18.

So we have this system of equations:

, 

Can you solve that system?  If not post again asking how.

Answer: Isaac is 28, Ruth is 19, and Brian is 10.

Checking:

Isaac is 28, and so he is nine years older than Ruth, who is 19.
That checks.

Isaac is 28, and is four times as old as Brian was three years ago.
Since Brian is 10 now, 3 years ago he was 7, and 28 is four times 7.
So that checks. 

Brian is 10, and therefore eighteen years younger than Issac who is
28.

So that answer is correct

Edwin