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: