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To find the distance between two parallel lines, we can use the formula for the distance between two lines in the slope-intercept form \( y = mx + b \), where the slope \( m \) is the same for both lines:

\[
\text{Distance} = \frac{|b_2 - b_1|}{\sqrt{1 + m^2}}
\]

For the given lines:
- \( y = x - 6 \) (slope \( m = 1 \), intercept \( b_1 = -6 \))
- \( y = x + 8 \) (slope \( m = 1 \), intercept \( b_2 = 8 \))

Let's apply the formula:

\[
\text{Distance} = \frac{|8 - (-6)|}{\sqrt{1 + 1^2}} = \frac{|8 + 6|}{\sqrt{2}} = \frac{14}{\sqrt{2}} = \frac{14\sqrt{2}}{2} = 7\sqrt{2}
\]

Thus, the distance between the two parallel lines is \( 7\sqrt{2} \).

Would you like further details or have any questions? Here are five related questions you might find interesting:

1. How do you derive the formula for the distance between two parallel lines?
2. What happens if the slopes of the lines are not the same?
3. Can you find the distance between lines using different forms, such as point-slope form?
4. What is the geometric significance of the distance formula between parallel lines?
5. How would the distance change if one of the lines was shifted vertically?

**Tip**: When finding the distance between parallel lines, always ensure that their slopes are identical.

Find the distance between the parallel lines y = x - 6 and y = x + 8.

Learn how to calculate the distance between two parallel lines y = x - 6 and y = x + 8. This detailed guide covers key concepts in analytic geometry, including the formula for the distance between two parallel lines, and is suitable for high school students in grades 9-12.

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