Since this linear system contains irrational numbers, how would you solve it "precisely"? How would the answer be any different than if a calculator was used? Do I just leave alone the irrational square root terms and express the answer like that? Thanks for any help.

Maybe the professor is trying to show you what happens when you solve a system like that with a calculator. I don't know what happens when you try to solve that with a calculator, but since the above works for all x,y in R I would guess that the calculator can't represent this solution or something. The professor might therefore be trying to show you the dangers of using a calculator to solve systems of equations. Just a guess :/

EDIT: Why it works for all x,y in R--multiply top by -sqrt(3)

If you multiply the first equation by sqrt(3), you will see that the two become identical. This means that they are linearly dependant and there will be NO unique solution. There will actually be an infinite number of solutions. Maybe that is what your prof was trying to show.

Parth Dave, so what about the calculator part? Would it hold true for the calculator part too? How would I know?

Also, if I had problem like:

[4*sqrt of 3 5 : 2 ]
[5 sqrt of 13 : 1 ]

What approach do I need to take here (same subject on solving precisely and solving by calculator)? I have to solve using determinants. He never covered this in class yet it's on the homework.

I tried that with the precise method by leaving alone the irrational numbers (keeping the square roots the way they are). And I end up with this really really long term that just doesn't seem right you know? Thanks for your help though.