Well, if you are taking a multiple choice test, then hopefully one (and only one) of the four answers works. This leads to a pragmatic (if slightly underhanded) approach to solving these problems, namely test all the values. This is likely to be faster than actually calculating $x$ and $y$ by hand if the number of choices is not too large.
Here's how you could do this problem with this approach. First, it seems easier to calculate $3x - y$ than $5x-3y$, so let's plug the 5 cases into the first equation.
a. $3x-y=3\times20-30=60-30=30$ works
b. $3x-y=3\times30-20=90-20=70$ no
c. $3x-y=3\times20-40=60-40=20$ no
d. $3x-y=3\times10-30=30-30=0$ no
e. $3x-y=3\times30-30=90-30=60$ no
So we've already ruled out all the answers other than (a), and hence you can just pick (a). We don't need the second equation at all.
Another possible closely related method (if you're pressed for time) would be after seeing that (a) works for the first equation, immediately check (a) against the second equation. If it works there too, you've found a correct answer, and there should only be one correct answer if you trust whoever wrote the exam, so you can just mark down (a). This is likely to save you time in a case like this where you quickly find one that satisfies the first equation.
As for understanding how to solve this outside such a controlled environment, amWhy's answer is very good as a basic treatment, so I won't repeat that. The basic idea is that you want to find a way to eliminate one variable from the equations by manipulating to solve for one variable and substituting into the other equation. Then solve that equation for the second variable, and go back to find the first variable after that.
If you want to do further study, this is a system of linear equations, and finding the solutions to such systems is a significant part of linear algebra. You probably don't want to try to just pick up a linear algebra textbook without a good understanding of solving systems like the above one though. I recommend trying the techniques above on a system with 3 equations and 3 unknowns to check that you understand how it works first.