Usually we call each line an "equation". What you have is a "system of linear equations". Since you also have some restrictions that look like linear inequalities ("all the numbers have to be positive"), you have yourself a linear programming
problem. Linear programming problems typically involve some objective function that has to be minimized or maximized. If you just want to find a solution, but don't care which one, you only have to use the part of the solver that finds an admissible solution, which is typically the first step in a linear-programming algorithm.
EDIT: In some sense you can think of this problem as finding a convex hull described as an intersection of hyperspaces. Chapter 15 of this book
seems to deal precisely with this problem.