Number of integer solutions to the equation $x_1+x_2+x_3+x_4=100$

Question

$x_1+x_2+x_3+x_4 = 100$ with

$1 \le x_1\le10$

$2\le x_2\le15$

$x_3\ge5$

$0\le x_4\le10$

Apparently this is the same as

$y_1 + y_2 + y_3 + y_4 = 92$ with

$y_1 \le 9$

$y_2 \le13$

$y_4 \le10$

I understand the $3$ conditions, but what has happened to $x_3$? I can see that it has been subtracted but what has happened to the conditions?

Answer 1

We are defining $y_1=x_1-1, y_2=x_2-2,y_3=x_3-5,y_4=x_4$ and demanding that all the $y$'s be $\ge 0$. Once we defined $y_3=x_3-5$, there is no constraint on $y_3$ except that it be nonnegative, so it is not listed in the constraints.