To find the equation of a line parallel to the given line $7x - 8y = -8$ and passing through the point $(1, 6)$, follow these steps:

1. Identify the slope of the given line

The equation $7x - 8y = -8$ can be rewritten in slope-intercept form ($y = mx + b$) by solving for $y$:

$7x - 8y = -8$

Subtract $7x$ from both sides:

$-8y = -7x - 8$

Divide by $-8$:

$y = \frac{7}{8}x + 1$

The slope $m$ of the given line is $\frac{7}{8}$.

2. Use the slope of the parallel line

Since parallel lines have the same slope, the slope of the new line is also $m = \frac{7}{8}$.

3. Use the point-slope form of the equation

The point-slope form is:

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

Substituting the point $(1, 6)$ and the slope $m = \frac{7}{8}$ into this form:

$y - 6 = \frac{7}{8}(x - 1)$

4. Simplify to the slope-intercept form

Expand and simplify the equation:

$y - 6 = \frac{7}{8}x - \frac{7}{8}$

Add 6 to both sides:

$y = \frac{7}{8}x - \frac{7}{8} + 6$

Convert 6 to a fraction with a denominator of 8:

$y = \frac{7}{8}x - \frac{7}{8} + \frac{48}{8}$

Simplify:

$y = \frac{7}{8}x + \frac{41}{8}$

Final Answer:

The equation of the line parallel to $7x - 8y = -8$ and passing through the point $(1, 6)$ is:

$y = \frac{7}{8}x + \frac{41}{8}$

Would you like more details on this solution or have any questions?

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Here are some related questions you might find interesting:

1. How do you find the equation of a line perpendicular to a given line?
2. How can you determine if two lines are parallel or perpendicular just by looking at their equations?
3. What is the general form of the equation of a line, and how does it differ from the slope-intercept form?
4. How do you find the intersection point of two lines given their equations?
5. What are the steps to convert between different forms of a linear equation?

Tip: Remember, parallel lines have identical slopes, while perpendicular lines have slopes that are negative reciprocals of each other.