x + y - z = -1

2x + 3y + tz = 3

x + ty + 3z = 2

1*3*3 + 2*t*(-1) + 1*1*t - 1*3*(-1) - 1*t*t - 3*2*1 =

9 - 2t + t + 3 - t^2 -6 = -t^2 -t +6

t^2 + t - 6 = 0 has one Real, Positive Solution and one Real Negative Solution. This is sufficient to state the system has a solution if you avoid those two places.