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?
Best Answer
Because summing gives $$4=0,$$ which is wrong.