Suppose we have a knapsack of size $K = 10$, and items of size
$x_1=4; x_2 = 5; x_3 = 6$. Clearly if $K - x_1 - x_3 = 0$, then set $A = 2K - x_1 - x_2 - x_3$ and $A - x_1 + x_2 - x_3 = 0$.

If $K$ is the size of a knapsack and $x_i$ is the weight of the $i$th item, 
then set $a_1 = 2K - \sum_{i} x_i$ and $a_i = x_{i-1}$. If you can determine if the expression $a_1 \pm a_2 \pm a_3 \pm \ldots$ is ever 0 you can solve the knapsack problem.