I am looking for a general way to find a solution to a system of equations in GF(2) such that the solution has the least amount of true variables. After Gaussian elimination I get a matrix like such:
$$ \left[ \begin{array}{ccccccccc|c}1&0&0&0&1&1&1&0&0&1\\ 0&1&0&0&1&0&1&0&1&1\\ 0&0&1&0&0&1&1&1&0&0\\ 0&0&0&1&1&1&1&1&1&1\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ \end{array} \right] $$
Finding a solution is trivial. If the variables are named $a$, $b$, ..., then $\{a, b, d\} = 1$ is a valid solution. However, $\{e\} = 1$ is also a valid solution, and only uses a single true variable.
In this example we got 'lucky' because it's easy to spot the minimal solution, but this obviously doesn't generalize. Is there a fast general method to find a solution that minimizes the number of true variables?