consider the line 7x-8y=-8 find the equation of the line that is parallel to this line and passes through the point (1,6)

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.

Learn how to find the equation of a line parallel to 7x - 8y = -8 that passes through the point (1, 6). This problem involves linear equations, parallel lines, point-slope form, and slope-intercept form. Suitable for Grades 7-9 students.

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