Here's a messy way.

Solve the 4 x 4 system you get by setting the 5th variable to 0 and only considering the first 4 equations.

Also solve the 4 x 4 system you get by setting the 5th variable to 1 and all the constants to 0 and only considering the first 4 equations.

If the first has solution (a1,a2,a3,a4) and the second (b1,b2,b3,b4), then solutions to the first 4 equations will be of the form (a1+t\*b1,a2+t\*b2,a3+t\*b3,a4+t\*b4,t).

Use the 5th equation to solve for t.

Edit: it's possible the first step won't have a solution, if so try a different variable in the role of the 5th variable.