jmc

Let Ted's age be $t$ and Sally's age be $s$. We are trying to find the value of $t$. We can write a system of two equations to represent the given information. Here are our two equations:

$$\begin{array}{rl}t& =2s-15\\ t+s& =54\end{array}$$The first equation represents the statement ``Ted's age is 15 years less than twice Sally's age.'' The second equation represents the statement ``The sum of their ages is 54.'' We are solving for $t$, so we want to eliminate $s$. From the second equation, we get that $s=54-t$. Substituting that into the first equation to get rid of $s$, we have $t=2(54-t)-15$, from which we get that $t=31$. Thus, Ted's age is $\boxed{31}$ years old.