To determine the slope of line $d$, which is parallel to line $c$, we only need the slope of line $c$.

From the equation of line $c$: $y = \frac{-8}{9}x + 9$

The slope of line $c$ is the coefficient of $x$, which is $\frac{-8}{9}$.

Since parallel lines have the same slope, the slope of line $d$ is also: $\boxed{-\frac{8}{9}}$

Would you like further clarification or more examples of finding slopes for parallel or perpendicular lines? Here are some related questions to explore:

1. How do you find the slope of a line perpendicular to a given line?
2. Can you explain why parallel lines always have the same slope?
3. What happens to the slope if the line equation is in standard form instead of slope-intercept form?
4. Can we calculate the slope if we’re given two points on a line instead of the equation?
5. How does the slope affect the steepness and direction of a line?

Tip: Remember that the slope is a measure of steepness; a negative slope indicates a line falling as it moves from left to right.