Excel found a solution to this but it's not pretty: (Be sure to remove the constraint on nonnegative integers)
To set up Solver you want to set x1 through x6 as inputs, set the sum of each combination as one half of the constraints and the numbers (-4,9,0) as the other half. Then choose one sum as the target cell to equal its equivalent right half (-4,9,0). I chose -4. Set your constraints to have your respective sums equal its right halves. Unclick the constraint for nonnegative integers and Solve! Be prepared for some ugly negative integers. Please attach a picture of your Solver setup if you have questions.
Hope that helps,