The answer is yes. Let's consider the following system of equations:
$$ which is equal to this one (after doing some manipulations on equations):
$$ The system is consistent but in second system the number of variables is more than the numbers of equations so the system has infinitely many solutions. Let $y$ and $w$ be our free variables and set $w=b$ and $y=a$ to find out if we can find the general solutions. According to last system we get $z=1+2b$ and $x=4-2a+b$ so $$(4-2a+b,a,1+2b,b)$$. Indeed, if you gave us the system you had in mind in which the third equation was $0=0$, then we would find that possible line.