To find the slope of the secant line $PQ$, we use the formula for the slope between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$, which is:

$\text{slope of PQ} = \frac{y_2 - y_1}{x_2 - x_1}$

Here, the point $P(6, 1)$ lies on the curve $y = x - 5$, meaning the y-coordinate of point P is $6 - 5 = 1$.

For the point $Q(x, y)$, the y-coordinate is given by $y = x - 5$. So, the formula for the slope becomes:

$\text{slope of PQ} = \frac{(x - 5) - 1}{x - 6} = \frac{x - 5 - 1}{x - 6} = \frac{x - 6}{x - 6} = 1$

However, this is valid only when $x \neq 6$. At $x = 6$, we would need to compute a limit or approach the slope using values close to 6. This is why we compute the slope for various values of $x$ as you’ve requested.

Let’s now compute the slope for the given values of $x$:

Values to compute:

1. $x = 5.5$
2. $x = 5.9$
3. $x = 5.99$
4. $x = 5.999$
5. $x = 6.5$
6. $x = 6.1$
7. $x = 6.01$
8. $x = 6.001$

Let me calculate the slopes for all these values.For all the given values of $x$, the slope of the secant line $PQ$ is:

1. For $x = 5.5$, the slope is $1.0$
2. For $x = 5.9$, the slope is $1.0$
3. For $x = 5.99$, the slope is $1.0$
4. For $x = 5.999$, the slope is $1.0$
5. For $x = 6.5$, the slope is $1.0$
6. For $x = 6.1$, the slope is $1.0$
7. For $x = 6.01$, the slope is $1.0$
8. For $x = 6.001$, the slope is $1.0$

In this case, the slope is consistently 1.0 for all points near $x = 6$, indicating that the secant lines are almost horizontal.

Would you like more details on this, or do you have any other questions?

Here are 5 related questions:

1. What happens to the slope as $x$ approaches 6?
2. How can you determine the equation of a tangent line at $x = 6$?
3. Can you find the derivative of the function $y = x - 5$?
4. How would the slope change if the curve were more complex, such as $y = x^2 - 5$?
5. How does this method extend to finding slopes of secant lines in 3D space?

Tip: The slope of a secant line between two points on a curve provides an average rate of change, while the derivative gives the instantaneous rate of change at a specific point.