jmc

We might try to factor the quadratic, but unfortunately that doesn't get us anywhere. Instead, we start by subtracting $5t$ from both sides so as to isolate the fraction. This gives us $$\frac{13t^2-34t+12}{3t-2}=t-1.$$Multiplying both sides by $3t-2$ gives $$13t^2-34t+12=(t-1)(3t-2).$$Expanding the right side gives $13t^2-34t+12=3t^2-5t+2$, so $10t^2-29t+10=0$. Factoring gives $(2t-5)(5t-2)=0$, which has solutions $t=2/5$ and $t=5/2$. The largest of these is $\boxed{\frac{5}{2}}$.