How to show that a system of linear equations is not solvable given certain constraints?

Imagine you have the following system of linear equations,

$$1 - 3x_1 - x_2 + x_3 + 3x_4 = 0$$ $$1 - x_1 - 5x_2 + 2x_3 + 4x_4 = 0$$ $$1 + x_1 + 2x_2 - x_3 - 2x_4 = 0$$ $$1 + 3x_1 + 4x_2 - 2x_3 - 5x_4 = 0,$$ and the constraints that $$x_1, x_2, x_3, x_4 \ge 0$$ and that $$x_1 + x_2 = x_3 + x_4.$$ I want to show that this system of linear equations is not solvable given the constraints, especially by using the second constraint. So for example, one could calculate $$(I) - (III)$$, which yields the equation $$-4x_1 - 3x_2 + 2x_3 + 5x_4 = 0,$$ which is equivalent to $$4x_1 + 3x_2 = 2x_3 + 5x_4.$$

Obviously, comparing the coefficients yields that this does not fullfill the second constraint. The only possibility would be that $$x_1 = x_2 = x_3 = x_4 = 0,$$ but in that case, we'd have $$1 - 0 = 0,$$ which is obviously false. Is this enough to show that the system can't be solved given the constraints?

Because summing gives $$4=0,$$ which is wrong.
• @Borol Linear programming duality provides a certificate of infeasibility in the form of dual multipliers for the constraints. In Michael's answer, "summing" corresponds to dual multipliers $(1,1,1,1)$. – RobPratt Sep 14 '20 at 13:31