I am given a system of (in)equalities (see below). There is a continuum of possible solutions and I just have to find one of them. My question is whether there is a faster way to find one of the solutions than solving the whole system? In particular, is there a faster method or can the system be solved faster with a "trick" I don't see? (I have to solve it by hand, so no programming possible)
$\frac{t_1^i}{3} + \frac{t_2^i}{3} + \frac{t_3^i}{3} = 0$
$\frac{t_4^i}{3} + \frac{t_5^i}{3} + \frac{t_6^i}{3} > 0$
$\frac{t_7^i}{3} + \frac{t_8^i}{3} + \frac{t_9^i}{3} > 0$
$\frac{t_4^i}{5} + \frac{t_5^i}{5} + \frac{3t_6^i}{5} = 0$
$\frac{t_1^i}{5} + \frac{t_2^i}{5} + \frac{3t_3^i}{5} > 0$
$\frac{t_7^i}{5} + \frac{t_8^i}{5} + \frac{3t_9^i}{5} > 0$
$\frac{t_7^i}{2} + \frac{t_8^i}{4} + \frac{t_9^i}{4} = 0$
$\frac{t_1^i}{2} + \frac{t_2^i}{4} + \frac{t_3^i}{4} > 0$
$\frac{t_4^i}{2} + \frac{t_5^i}{4} + \frac{t_6^i}{4} > 0$