Find Variable Cost per Unit
Based on the information you have given, both the solution manual and your solution are wrong. You are correct that the fixed cost for system I is 1. Variable cost is given by $$VC(x)=TC(x)-FC.$$ The variable cost for system I is therefore equal to $0.9x$. Note that the variable cost depends on $x$! For system II the variable cost is $0.1x$.

You appear to have confused variable cost with marginal cost. Variable cost is the part of the total cost that changes with quantity; marginal cost is, roughly, the cost of producing one more unit (and is given by the slope of the total cost curve). Thus, the marginal cost of system I is $0.9$ and the marginal cost of system II is $0.1$.