To solve this problem, we need to find the equation of a line that passes through the point $\left( \frac{1}{8}, -1 \right)$ and is parallel to the line given by $8x - 9y = 8$.

Step 1: Find the slope of the given line

The equation of the given line is in standard form: $8x - 9y = 8$. First, we convert it to slope-intercept form $y = mx + b$, where $m$ is the slope.

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

From this equation, we can see that the slope of the line is $m = \frac{8}{9}$.

Step 2: Use the slope for the parallel line

Since the new line is parallel to the given line, it will have the same slope, $m = \frac{8}{9}$.

Step 3: Use the point-slope form to find the equation

Now, we use the point-slope form of a line, which is: $y - y_1 = m(x - x_1)$ Substitute $m = \frac{8}{9}$, $x_1 = \frac{1}{8}$, and $y_1 = -1$:

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

Simplify: $y + 1 = \frac{8}{9}x - \frac{8}{9} \times \frac{1}{8}$ $y + 1 = \frac{8}{9}x - \frac{1}{9}$ Subtract 1 from both sides: $y = \frac{8}{9}x - \frac{1}{9} - 1$ $y = \frac{8}{9}x - \frac{1}{9} - \frac{9}{9}$ $y = \frac{8}{9}x - \frac{10}{9}$

Final Answer:

The equation of the line is: $y(x) = \frac{8}{9}x - \frac{10}{9}$

Would you like further details or explanations?

Here are 5 related questions to explore:

1. How do you convert between standard form and slope-intercept form of a line?
2. What are the steps to find the equation of a line perpendicular to a given line?
3. How can you find the distance between a point and a line?
4. How do you find the equation of a line given two points?
5. What is the significance of the slope in parallel and perpendicular lines?

Tip: When finding the equation of a parallel line, the key step is identifying that the slope must remain the same.