To find the slope of a line passing through two points, we use the slope formula:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Given points \( (1, 6 \frac{1}{2}) \) and \( (6, 38 \frac{5}{8}) \):

1. Convert the mixed numbers to improper fractions:
   - \( 6 \frac{1}{2} = \frac{13}{2} \)
   - \( 38 \frac{5}{8} = \frac{309}{8} \)

2. Apply the slope formula:

\[
\text{slope} = \frac{\frac{309}{8} - \frac{13}{2}}{6 - 1}
\]

First, find the difference in \( y \)-values:

\[
\frac{309}{8} - \frac{13}{2} = \frac{309}{8} - \frac{52}{8} = \frac{309 - 52}{8} = \frac{257}{8}
\]

Now, divide by the difference in \( x \)-values (which is \( 6 - 1 = 5 \)):

\[
\text{slope} = \frac{\frac{257}{8}}{5} = \frac{257}{8 \times 5} = \frac{257}{40}
\]

So, the slope is \( \frac{257}{40} \).

Would you like more details or have any questions?

Here are five related questions you could explore:
1. How do you find the equation of the line using the slope and a point?
2. Can you simplify improper fractions in slope calculations?
3. What is the significance of a positive slope in a real-world context?
4. How do you convert between different forms of slope-intercept equations?
5. What are the applications of slope in calculus?

**Tip:** When working with mixed numbers, always convert them to improper fractions to avoid mistakes during calculations!

Learn how to calculate the slope of a line passing through points (1, 6 1/2) and (6, 38 5/8) with a step-by-step solution. This problem covers key geometry concepts, including the slope formula and working with mixed numbers. Suitable for students in Grades 8-10.

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