When you have $\dfrac{1}{3} - \dfrac{1}{x-2}$, find a common denominator (involving both 3 and $x-2$) and be careful with subtraction in the numerator.

Once you have an equation of the form $\dfrac{1}{y-2} = \text{(fraction in }x\text{)}$, think about how taking reciprocals on both sides helps you get an expression for $y-2$.

After you find $y-2$ in terms of $x$, add 2 to both sides and simplify the resulting complex fraction so it looks like one of the answer choices.

In Desmos, keep the original relationship in mind: $1/(x-2) + 1/(y-2) = 1/3$. For each answer choice, you'll temporarily assume that $y$ is given by that choice and see whether it satisfies the equation.

For a given option (say option A), define a function for $y$ in terms of $x$, for example yA = (5x - 16)/(x - 5). Then create a new expression representing the left side minus the right side of the original equation using that $y$: fA(x) = 1/(x-2) + 1/(yA - 2) - 1/3.

Graph the function you just defined (e.g., fA(x)). If that graph lies exactly on the x-axis (i.e., $fA(x) = 0$) for all $x$ in its domain (excluding places where denominators are zero), then that choice satisfies the original equation for all such $x$. If the graph is not identically on the x-axis, that option is incorrect.

Repeat the same process for options B, C, and D: define yB, yC, yD in terms of $x$, build the corresponding fB(x), fC(x), fD(x), and graph them. The option whose function stays on the x-axis (aside from vertical asymptotes) is the correct expression for $y$ in terms of $x$.

Start from the given equation:

Subtract $\dfrac{1}{x-2}$ from both sides to isolate the term with $y$:

Now the right-hand side is an expression in terms of $x$ only.

To combine $\dfrac{1}{3}$ and $\dfrac{1}{x-2}$, use the common denominator $3(x-2)$.

Rewrite each fraction:

• $\dfrac{1}{3} = \dfrac{x-2}{3(x-2)}$
• $\dfrac{1}{x-2} = \dfrac{3}{3(x-2)}$

So

Thus,

We have

Since both sides are nonzero fractions, take the reciprocal of both sides:

Now $y$ is almost isolated; one more step will express $y$ entirely in terms of $x$.

From

add $2$ to both sides. To combine $2$ with the fraction, write $2$ with denominator $x-5$:

Combine the numerators:

So the correct equation expressing $y$ in terms of $x$ is $y = \dfrac{5x - 16}{x - 5}$, which matches choice A.