Determine the slope of the tangent line, then find the equation of the tangent line at $t=-1$.
\[
x=6 t, y=t^{7}
\]
Slope: $\frac{7}{6}$
Equation:

Step 1 ：We are given the parametric equations \(x=6t\) and \(y=t^{7}\). We are asked to find the slope of the tangent line and the equation of the tangent line at \(t=-1\).

Step 2 ：First, we need to find the derivative of the function. The derivative of a function at a certain point gives the slope of the tangent line at that point. In this case, we have parametric equations, so we need to find \(\frac{dy}{dx}\). We can find \(\frac{dy}{dx}\) by finding \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\) and then dividing \(\frac{dy}{dt}\) by \(\frac{dx}{dt}\).

Step 3 ：For \(x=6t\), \(\frac{dx}{dt}=6\). For \(y=t^{7}\), \(\frac{dy}{dt}=7t^{6}\). Therefore, the slope of the tangent line is \(\frac{7t^{6}}{6}\).

Step 4 ：At \(t=-1\), the slope of the tangent line is \(\frac{7(-1)^{6}}{6}=\frac{7}{6}\).

Step 5 ：To find the equation of the tangent line, we can use the point-slope form of a line, which is \(y - y_{1} = m(x - x_{1})\), where m is the slope and \((x_{1}, y_{1})\) is a point on the line. In this case, the point is given by the values of x and y at \(t=-1\).

Step 6 ：At \(t=-1\), \(x=-6\) and \(y=-1\). Therefore, the equation of the tangent line is \(y - (-1) = \frac{7}{6}(x - (-6))\), which simplifies to \(y = \frac{7}{6}x + \frac{7}{6}\).

Step 7 ：\(\boxed{\text{The slope of the tangent line at } t=-1 \text{ is } \frac{7}{6} \text{ and the equation of the tangent line at } t=-1 \text{ is } y = \frac{7}{6}x + \frac{7}{6}}\)