Why at the last step, when we have found optimal basic feasible solution which implies also optimal corner point feasible solution, the right-hand values of equations of system (augmented form of constraints) actually coincide with final values of basic variables? For example let
$Z = 3x_1+5x_2\to max$
$x_1 \le 4$
$2x_2 \le 12$
$3x_1+2x_2 \le 18$
At the last step of solving we'll get $Z = 36 - \frac{3}{2}x_4 - x_5$. So as all coefficients are negative then it is the most optimal solution. And from here we can see that $x_4$ and $x_5$ are non-basic variables at this step. While $x_1, x_2, x_3$ are basic. And equations will be in the form:
$x_3 + \frac{1}{3}x_4 - \frac{1}{3}x_5 = 2$
$x_2 + \frac{1}{2}x_4 = 6 $
$x_1 - \frac{1}{3}x_4 + \frac{1}{3}x_5 = 2$
And the final solution for the problem is $(2; 6; 2; 0; 0)$. You can see that all values in solution coincide with values of equations. Can you please explain me, why is it happpening? What is underlying logic here?