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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?

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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.

Find a linear equation whose graph is the straight line through (1/8, -1) and parallel to the line 8x - 9y = 8.

Learn how to find the equation of a line that passes through the point (1/8, -1) and is parallel to the line 8x - 9y = 8. The solution uses slope-intercept form and point-slope form to find the equation of the parallel line. Suitable for students in Grades 9-11.

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