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We are tasked with finding the equation of line \( s \), which is parallel to line \( r \) and passes through the point \( (8, 8) \). Line \( r \)'s equation is given as:

\[
y - 6 = \frac{1}{4}(x - 8)
\]

### Step 1: Find the slope of line \( r \)
Line \( r \)'s equation is in point-slope form: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope. From the equation:

\[
y - 6 = \frac{1}{4}(x - 8)
\]

The slope \( m \) is \( \frac{1}{4} \).

### Step 2: Use the slope for line \( s \)
Since line \( s \) is parallel to line \( r \), it has the same slope. Therefore, the slope of line \( s \) is also \( \frac{1}{4} \).

### Step 3: Apply the point-slope form to line \( s \)
We use the point-slope form of the equation again, but now for line \( s \), which passes through the point \( (8, 8) \):

\[
y - y_1 = m(x - x_1)
\]

Substitute \( m = \frac{1}{4} \), \( x_1 = 8 \), and \( y_1 = 8 \):

\[
y - 8 = \frac{1}{4}(x - 8)
\]

### Step 4: Convert to slope-intercept form
Now, simplify this equation to get it into slope-intercept form (\( y = mx + b \)):

\[
y - 8 = \frac{1}{4}(x - 8)
\]

Distribute \( \frac{1}{4} \) on the right side:

\[
y - 8 = \frac{1}{4}x - 2
\]

Add 8 to both sides:

\[
y = \frac{1}{4}x + 6
\]

### Final Equation:
The equation of line \( s \) in slope-intercept form is:

\[
y = \frac{1}{4}x + 6
\]

Would you like more details or have any further questions?

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Here are five related questions to explore further:
1. What is the general form of the equation of a line?
2. How can we determine if two lines are perpendicular?
3. What is the relationship between the slopes of parallel lines?
4. How do we find the equation of a line given two points?
5. What does the slope of a line represent geometrically?

**Tip**: When lines are parallel, they always have the same slope, no matter what point they pass through.

The equation of line r is y - 6 = 1/4(x - 8). Line s is parallel to line r and passes through (8, 8). What is the equation of line s?

Learn how to find the equation of a line parallel to y - 6 = 1/4(x - 8) that passes through (8, 8). This problem explores linear equations, the slope-intercept form, and the properties of parallel lines. Suitable for students in Grades 8-10.

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