You are trying to find relationships between s and t. That is very difficult. Instead, try to find relationships between x and y. What do you get from $\displaystyle x^2+y^2$ (use the first parametric equation)? What about (x-2)^2+(y-3)^2 (use the second parametric equation)?
Thanks, Slip! Using the cos^2+sin^2=1 theorem I ended up with
4 cos(s) = -7 -6 sin (s)
So x= 2 + -7 - 6 sin(s)
And adding that to the other equation, y=3+4sin(s), I ended up with
6y + 4x = -2
And from this I can get y= -2/3 x -1/3
So is the slope -2/3? I took an intuitive approach and aren't 100% sure if I did it right