Estimate integer solutions to a set of linear or diophantine equations (underdetermined)

Find a possible solution for every team's individual match score. Given n teams and m matches, there are n*m variables, or columns to solve for.

Each match has 3 teams. The total score for each match is known.

Each team has 12 matches. The total score for each team is known.

The total number of teams and matches is known, and the answers are all integers.

first, I broke the problem into 2 parts.

Match equations

Since there are 3 teams in a match, I gave teams competing in the match a coefficient of 1, and teams that did not compete in the match a coefficient of 0. The solutions to this matrix are the scores for each match.

team equations

Since each team competes in 12 matches, matches the team competed in had a coefficient of 1, and otherwise 0. The solutions to this matrix are the team score totals.

I then combined the 2 matrixes into the matrix to solve. This matrix has matches + teams rows, and matches * teams columns.

I tried using the least-squares solution, but this returns decimals, which my answers are not.

My question is, how can I find a possible answer to the system for solutions? Is there a linear diophantine matrix algorithm or tool to use?

More background on the question can be found in this question.