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The given curve is:

\[
y = \frac{x + 1}{x - 7}
\]

You are asked to:

1. **Find the slope \(m\) of the tangent line** at the point \((6, -7)\).
2. **Find the equation of the tangent line** at that point.

### Step 1: Find the slope \(m\)

To find the slope of the tangent line, you need the derivative of the function. The function \( y = \frac{x + 1}{x - 7} \) is a quotient, so we will apply the quotient rule.

The **quotient rule** states:
\[
\left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2}
\]

For this problem:
- \( u(x) = x + 1 \)
- \( v(x) = x - 7 \)

Now, compute the derivatives:
- \( u'(x) = 1 \)
- \( v'(x) = 1 \)

Using the quotient rule:
\[
y' = \frac{(x - 7)(1) - (x + 1)(1)}{(x - 7)^2}
= \frac{x - 7 - (x + 1)}{(x - 7)^2}
= \frac{x - 7 - x - 1}{(x - 7)^2}
= \frac{-8}{(x - 7)^2}
\]

Thus, the derivative is:
\[
y' = \frac{-8}{(x - 7)^2}
\]

To find the slope at the point \( (6, -7) \), plug \( x = 6 \) into the derivative:
\[
y'(6) = \frac{-8}{(6 - 7)^2} = \frac{-8}{(-1)^2} = -8
\]

So, the slope of the tangent line at \( x = 6 \) is:
\[
m = -8
\]

### Step 2: Find the equation of the tangent line

The equation of the tangent line is given by the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope, and \( (x_1, y_1) \) is the point of tangency. Here, \( m = -8 \), \( x_1 = 6 \), and \( y_1 = -7 \).

Substitute the values:
\[
y - (-7) = -8(x - 6)
\]
\[
y + 7 = -8(x - 6)
\]
Simplify:
\[
y + 7 = -8x + 48
\]
\[
y = -8x + 48 - 7
\]
\[
y = -8x + 41
\]

### Final Answer:
1. The slope \( m = -8 \).
2. The equation of the tangent line is \( y = -8x + 41 \).

Let me know if you'd like more details or have any questions!

Here are some related questions:
1. How do you apply the quotient rule in other situations?
2. What happens if you compute the derivative using a different method, like product rule?
3. How would the equation change if the curve was shifted horizontally?
4. What if the point of tangency changes to another point?
5. How do you check if the tangent line actually touches the curve only at one point?

**Tip:** When using the quotient rule, carefully compute the numerator and denominator separately to avoid mistakes.

Consider the curve y = (x + 1) / (x - 7). Find the slope m of the tangent line at the point (6, -7). Find the equation of the tangent line to the curve at the point (6, -7).

This calculus problem involves finding the slope of the tangent line to the curve y = (x + 1) / (x - 7) at the point (6, -7) using the quotient rule, and determining the equation of the tangent line. It includes a detailed step-by-step solution using differentiation.

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