You have something of the form

• P = Q = R

From here, we can get two different equations:

1. P = Q
2. P = R

We could get another equation, Q = R, but that follows from the previous ones, so it doesn't matter.

In your case,

1. A + B + C = 10 A + B - C
2. 10 A + B - C = 3

This is a system of linear equations. The solutions are B = 3 - 11A/2 and C = 9A / 2. So any integer solution will have A as even. In fact, for any even number, there is a solution which has A as that even number. There are infinite even numbers, and thus, infinite integer solutions.

Example: I like the number 282, which is even. That means that B = 3 - 11A/2 = -1548 and C = 9A / 2 = 1269. So a solution to the problem would be A = 282, B = -1548 and C = 1269.