Given two lines to find their intersection.

Your approach is correct so far. Since equation 2 has a nice coefficient of $1$ in front of the $t$, we can use that to easily eliminate the $t$ from the two others:

$$ Eq_1 - 3Eq_2: \qquad\qquad 4s = -5 \\ Eq_3 + 4Eq_2: \qquad \qquad -9s = 9 $$ These require different values of $s$, so they can never be satisfied at the same time. So the system of equations has no solutions, and the two lines do not intersect.