eliminate the parameter to find the cartesian equation of the curve for x=e^t, y=e^-5t.
We want to eliminate the parameter t and write y in terms of x.
From the given equations, we have:
x = e^t
y = e^(-5t)
Taking the natural logarithm of both sides of the first equation, we get:
ln(x) = t
Substituting this expression for t in the second equation, we get:
y = e^(-5 ln(x))
y = x^(-5)
Therefore, the Cartesian equation of the curve is y = x^(-5).
or,
y = e^(-5t) = (e^t)^-5 = x^-5