filters - Basic “filtering” in digital communications

I am completely stuck on solving some problems for my digital communications class. Junior level undergrad class, so fairly basic. I have been searching around all day for related articles etc but can't seem to find any that are helpful. Any help would be much appreciated. Here are the three questions:

For the coefficients my initial hunch is just to make c2 and c3 = 0, which would leave $\hat{y}[1] = y[1]$. My other attempt is to create a system of equations so the $x[2]$ terms are cancelled and the $x[1]$ terms are the only ones left over, but this system proves difficult to me and I am left with a mess of noise. For some reason I can only get two equations with three unknowns.

$$c_3= -3 c_2 \\ c_1 + \frac13 c_2 + \frac14 c_3 = 1$$ where $c_1,c_2,c_3$ are constants.

Sorry for my terrible formatting. I believe I completed the first question correctly, here was my process.

Moving on to question 2, for part A I'm fairly sure it is what I was attempting to do before so I got c3=-3 * c2

for part b, I am stuck again because my brain keeps going back to the idea that I should just make c2 and c3 both 0, because it appears to me our goal is to make x1 recoverable.