After you clear the fraction, you will get an equation with both $2ut$ and $t$ in it. Move all terms involving $t$ to one side and everything else to the other side.

Once all $t$ terms are on one side, factor out $t$. Then divide both sides by the factor multiplying $t$ to isolate $t$ and simplify any negative signs carefully.

In Desmos, define a function that represents the right-hand side of the original equation in terms of $t$ (use a different variable like $x$ if you prefer):

• Type: f(u) = (t - 5) / (2t + 1) is hard to work with directly, so instead think of $t$ as a function of $u$.
• For checking, we will substitute each answer choice into the right-hand side and see if we get back $u$.

Pick one answer choice and define $t$ as a function of $u$ using that expression. For example, for a generic choice write:

• t(u) = [expression from the choice]

Then define what the original right-hand side becomes when using this $t(u)$:

• g(u) = ( t(u) - 5 ) / ( 2*t(u) + 1 )

Now also enter:

• h(u) = u

Look at the graphs of $g(u)$ and $h(u)$. If for a particular answer choice the graph of $g(u)$ lies exactly on top of the graph of $h(u)$ (a straight line through the origin with slope 1) over its domain (excluding any points where there is a vertical asymptote), then that choice correctly expresses $t$ in terms of $u$. Repeat this process for each choice to see which one works.

Start with the given equation:

Since $2t + 1 \ne 0$, multiply both sides by $2t + 1$ to eliminate the denominator:

Distribute $u$ on the left side:

We want to get all the $t$ terms together. Subtract $t$ from both sides:

Now group the $t$ terms and the constant terms:

Factor $t$ out of the left side:

Now factor a negative sign out of the right side to make it easier to see the pattern:

So the equation becomes:

Solve for $t$ by dividing both sides by $(2u - 1)$:

Now simplify the negative sign by multiplying numerator and denominator by $-1$:

This matches choice D: $t = \dfrac{5 + u}{1 - 2u}$.