Problem #PRU-78130

Problem

There is a system of equations $$\begin{array}{rl}\ast x+\ast y+\ast z& =0,\\ \ast x+\ast y+\ast z& =0,\\ \ast x+\ast y+\ast z& =0.\end{array}$$ Two people alternately enter a number instead of a star. Prove that the player that goes first can always ensure that the system has a non-zero solution.