Answer
The solution set of the rational equation $\frac{1}{2}+\frac{1}{3}=\frac{1}{t}$ is $\left\{ \frac{6}{5} \right\}$.
Work Step by Step
$\frac{1}{2}+\frac{1}{3}=\frac{1}{t}$
Note that, $t\ne 0$.
The denominators of the rational expressions are $2$, $3$, and $t$.
Multiply both sides by $6t$.
$6t\left( \frac{1}{2}+\frac{1}{3} \right)=6t\left( \frac{1}{t} \right)$
Apply the distributive property,
$6t\cdot \left( \frac{1}{2} \right)+6t\cdot \left( \frac{1}{3} \right)=6t\left( \frac{1}{t} \right)$
Simplify the terms,
$\begin{align}
& \frac{6t}{2}+\frac{6t}{3}=\frac{6t}{t} \\
& \frac{2\cdot 3\cdot t}{2}+\frac{2\cdot 3\cdot t}{3}=\frac{6t}{t} \\
& 3t+2t=6
\end{align}$
Again, apply the distributive property,
$\begin{align}
& 3t+2t=6 \\
& 5t=6
\end{align}$
Apply the Multiplication principle, divide both sides by 5:
$\begin{align}
& \frac{5t}{5}=\frac{6}{5} \\
& t=\frac{6}{5}
\end{align}$
Now, check the solution of the equation$\frac{1}{2}+\frac{1}{3}=\frac{1}{t}$
Substitute $t=\frac{6}{5}$ in $\frac{1}{2}+\frac{1}{3}=\frac{1}{t}$:
$\begin{align}
& \frac{1}{2}+\frac{1}{3}\overset{?}{\mathop{=}}\,\frac{1}{\left( \frac{6}{5} \right)} \\
& \frac{1}{2}+\frac{1}{3}\overset{?}{\mathop{=}}\,\frac{5}{6} \\
& \frac{3+2}{6}\overset{?}{\mathop{=}}\,\frac{5}{6} \\
& \text{ }\frac{5}{6}=\frac{5}{6} \\
\end{align}$
Both sides of above equation are same. So, $t=\frac{6}{5}$ is the solution of $\frac{1}{2}+\frac{1}{3}=\frac{1}{t}$.