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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.

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

Learn how to find the equation of a line that is parallel to 8x - 9y = 8 and passes through the point (1/8, -1). This step-by-step solution involves finding the slope and using point-slope form. Suitable for students in Grades 8-10.

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