Question

given the parametric equations below, eliminate the parameter t to obtain an equation for y as a function of x\

$$\begin{cases}x(t) = e^{5t}\\\\y(t) = e^{8t}\\end{cases}$$

\\(y(x) = \\)\boxed{?}question help: \\(\boldsymbol{\text{video}}\\)

Explanation:

Step1: Solve for \( t \) from \( x(t) \)

Given \( x = e^{5t} \), take the natural logarithm of both sides: \( \ln(x) = \ln(e^{5t}) \). Using the property \( \ln(e^a)=a \), we get \( \ln(x) = 5t \). Then solve for \( t \): \( t=\frac{\ln(x)}{5} \).

Step2: Substitute \( t \) into \( y(t) \)

Given \( y = e^{8t} \), substitute \( t=\frac{\ln(x)}{5} \) into it: \( y = e^{8\cdot\frac{\ln(x)}{5}} \). Using the exponent rule \( a^{bc}=(a^b)^c \) and \( e^{\ln(a)}=a \), we can rewrite this as \( y = (e^{\ln(x)})^{\frac{8}{5}} = x^{\frac{8}{5}} \).

Answer:

\( y(x)=x^{\frac{8}{5}} \)