There are lots of these kinds of puzzles here, in the alphametic tag. As for how to solve them, it's a matter of noticing relationships to rule out certain values, then trial and error, choosing values until you get a contradiction.
Take your simpler
MAKE + KEYS = PSST example. Since there is no "carry", M and K must total 9 or less, which rules out them both being 5 or above, and later, when you know one of them, puts upper bounds on the other. It also constrains P: if they are 4 and 5, P is 9. If they were 1 and 0, P would be 1, but can't be. So you can rule out P being 1 or 0 (two different positive numbers can't add to zero.)
Some people actually write out equations:
E + S = T or T + 10
K + Y (+ maybe 1) = S or S + 10
Then they either solve simultaneous equations or just plug in some guesses.
The multiplying one is harder, and uses different relationships. So you know that
T * M = E + 10 or 20 or ....
And then there are things like "ah that means this letter must be odd" to help continue to narrow down the field.
You mentioned a teacher; are you expected to write a program to do this, or do it by hand?