Step 1. First substitute $\displaystyle V_3$ in equations (2) and (4) with the expression given in the equation (3).
Step 2. Next, in the equation (5) express $\displaystyle V_2$. Use that expression as a substitution for $\displaystyle V_2$ in equation (4). Now equation (4) should represent $\displaystyle V_4$ in terms of z's, k's and Y(z).
Step 3. Next use the expression for $\displaystyle V_4$ in equation (4) as a substitution for $\displaystyle V_4$ in equation (1). Now equation (1) should give you the expression for $\displaystyle V_1$ in terms of z's,k's and X(z) and Y(z).
Now you have to deal with the equation (2). Substitute $\displaystyle V_1$ with the expression in the equation (1) from the step 3. On the left side of the equation term $\displaystyle V_2(z)$ substitute with the expression obtained from the equation (5) (see step 2.)
After that that equation should contain no V's and I presume thats the equation you've been chasing all along?