As galvatron notes, there are only 26 possible keys, so you can just try them all even if you're working by hand. (If you have a computer, it's trivial.) Note that you don't need to decrypt the entire ciphertext with each key; just decrypting, say, the first word should be enough to rule out most if not all wrong keys.
In this particular case, since your ciphertext has word breaks, you can also exploit the fact that all normal English words contain at least one vowel (including
Y). So, in particular, you can pretty safely assume that one of the two letters in the ciphertext word
DI must decrypt to a vowel, which limits the possible keys to just 2 × 6 = 12 (including the null key that encrypts
I, which we may immediately rule out). In fact, only one of these keys yields a common English word (
IN), and indeed turns out to be the correct one.
(Of course, in general, you might be unlucky and the "word" you picked might turn out to be an acronym with no vowels at all. If so, you'll notice it when none of the decryptions makes any sense, at which point you can go back and try the other keys you previously ruled out as unlikely.)
As galvatron also notes, a good general method for breaking simple substitution ciphers is frequency analysis. Basically, you start by counting the number of times each letter occurs in the ciphertext, and then assume that the most common letters in the ciphertext most likely correspond to the most common letters in plain English text.
Unfortunately, for your specific example ciphertext, frequency analysis is somewhat less effective than usual. Not only is the ciphertext too short to yield reliable frequency data (most letters in it occur only once), but it looks as if the plaintext might even have been deliberately chosen to have an untypical frequency distribution. Of the three most common English letters,
T is completely absent from the plaintext, and
A both occurs only once, in the word
EXAM, which also contains an
X (one of the rarest letters in English, after
That said, if you persist with frequency analysis, trying to match the most common ciphertext letter (
J, occurring 3 times) to the most common English letters (which can be easily memorized as the nonsense phrase ETAOIN SHRDLU), you're likely to hit the correct match (
O) after only three false starts. If the author of the question had really wanted to mess up frequency analysis, they'd have replaced the word