We are tasked with finding a linear equation of a line that passes through the point $\left(\frac{1}{8}, -1\right)$ and is parallel to the line $8x - 9y = 8$.

Step 1: Identify the slope of the given line.

The given line is $8x - 9y = 8$. To find its slope, we need to rewrite the equation in slope-intercept form $y = mx + b$, where $m$ is the slope.

Start by solving for $y$:

$8x - 9y = 8$ $-9y = -8x + 8$ $y = \frac{8}{9}x - \frac{8}{9}$

Thus, the slope of the given line is $m = \frac{8}{9}$.

Step 2: Use the point-slope form.

Since the new line is parallel to the given line, it will have the same slope. The equation of a line with slope $m$ passing through a point $(x_1, y_1)$ is given by the point-slope form:

$y - y_1 = m(x - x_1)$

Here, the point is $\left(\frac{1}{8}, -1\right)$ and the slope is $m = \frac{8}{9}$. Substituting these values into the equation:

$y - (-1) = \frac{8}{9}\left(x - \frac{1}{8}\right)$ $y + 1 = \frac{8}{9}x - \frac{8}{9} \times \frac{1}{8}$ $y + 1 = \frac{8}{9}x - \frac{1}{9}$

Step 3: Simplify the equation.

Now, subtract 1 from both sides to get the final equation:

$y = \frac{8}{9}x - \frac{1}{9} - 1$ $y = \frac{8}{9}x - \frac{1}{9} - \frac{9}{9}$ $y = \frac{8}{9}x - \frac{10}{9}$

Thus, the equation of the line is:

$y(x) = \frac{8}{9}x - \frac{10}{9}$

Let me know if you'd like further explanation or have any questions! Here are a few related questions:

1. How do you find the slope of a line from its general form?
2. What is the significance of parallel lines in terms of slope?
3. How would you find the equation of a line perpendicular to $8x - 9y = 8$?
4. Can you explain the steps involved in converting from standard form to slope-intercept form?
5. How would the equation change if the line passed through a different point but remained parallel?

Tip: Always check if the given line is in standard form; rewriting it in slope-intercept form will help you easily identify its slope.