To find the equation of a line passing through the point $\left(\frac{1}{8}, -1\right)$ and parallel to the line $8x - 9y = 8$, follow these steps:

Step 1: Determine the Slope of the Given Line

Since the line we want to find is parallel to $8x - 9y = 8$, it will have the same slope as this line.

1. Rewrite $8x - 9y = 8$ in slope-intercept form $y = mx + b$. $-9y = -8x + 8$ $y = \frac{8}{9}x - \frac{8}{9}$ Here, the slope $m$ of the given line is $\frac{8}{9}$.

Step 2: Use the Point-Slope Form

Since the desired line is parallel, it also has a slope of $\frac{8}{9}$. We use the point-slope form of a line equation: $y - y_1 = m(x - x_1)$ Substitute $m = \frac{8}{9}$ and the point $\left(\frac{1}{8}, -1\right)$: $y - (-1) = \frac{8}{9}\left(x - \frac{1}{8}\right)$ $y + 1 = \frac{8}{9}x - \frac{8}{72}$ $y = \frac{8}{9}x - \frac{1}{9} - 1$