Let $t$ be TNYWR.

Alida, Bono, and Cate each have some jelly beans.

The number of jelly beans that Alida and Bono have combined is $6 t+3$.

The number of jelly beans that Alida and Cate have combined is $4 t+5$.

The number of jelly beans that Bono and Cate have combined is $6 t$.

How many jelly beans does Bono have?

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Let $A$ be the number of beans that Alida has, $B$ be the number of beans that Bono has, and $C$ be the number of beans that Cate has.

The given information translates to

$$
\begin{aligned}
& A+B=6 t+3 \\
& A+C=4 t+5 \\
& B+C=6 t
\end{aligned}
$$

Adding these three equations together gives $2(A+B+C)=16 t+8$, which implies $A+B+C=8 t+4$.

Subtracting $A+C=4 t+5$ from $A+B+C=8 t+4$ gives

$$
B=(A+B+C)-(A+C)=(8 t+4)-(4 t+5)=4 t-1
$$

Substituting $t=4$ into $B=4 t-1$ gives $B=4(4)-1=15$.