<s>[INST] Find The Equation This is the question:
Find the equations of the tangent lines to the curve $y = x − \frac 1x + 1$ that are parallel to the line $x − 2y = 3$.
There are two answers: 1) smaller y-intercept; 2) larger y-intercept
The work:
The slope of the line is (1/2).
$y' = ((x + 1) - (x - 1))/(x + 1)^2 = 2/(x + 1)^2 = 1/2$
=> $(x + 1)^2 = 4$
=> $(x + 1) = 2$ and $ x + 1 = -2$
=> $x = 1$ and $x = -3$
At $x = 1$, $y = (x - 1)/(x + 1) = 0$
The equation of the tangent is $y/(x - 1) = (1/2)$
=> $2y = x - 1$
=> $x - 2y - 1 = 0$
At $x = -3$,$ y = (x - 1)/(x + 1) = -4/-2 = 2$
The equation of the tangent is $(y - 2)/(x + 3) = (1/2)$
=> $2y - 4 = x + 3$
=>$ x - 2y + 7 = 0$
 [/INST] Two lines are parallel if they have the same slope, and the slope of a tangent line to a curve is the value of the derivative of the curve's function at that point. Therefore, follow this strategy.
1) Find the slope of the given line.
2) Find the derivative of the function defining the curve.
3) Set those expressions equal to each other and solve. This gives you the $x$-coordinate(s) of the intersection point(s) of the curve and the desired tangent line(s).
4) Find the $y$-coordinate for each $x$-coordinate you got in step 3.
5) Use the slope (from step 1) and the point(s) (from step 3) to find the equation(s) of the tangent line(s).
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