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Your question is a little imprecise. In general, when the integers $x_i$'s are bounded above and below, such problems are very difficult to decide whether there is a solution see the Knapsack Problem, the Subset Sum Problem, and Integer Linear Programming.

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Your question is a little imprecise. In general, such problems are very difficult to decide whether there is a solution see the Knapsack Problem, the Subset Sum Problem, and Integer Linear Programming.