1. **State the problem:** We need to find how many contemporary DVD titles Jarred has given the system of equations: $$x + y = 3500$$ $$x - y = 2342$$ where $x$ is the number of contemporary titles and $y$ is the number of classic titles. 2. **Formula and method:** We will use the method of linear combinations (adding the two equations) to eliminate $y$ and solve for $x$. 3. **Add the two equations:** $$\begin{aligned} (x + y) + (x - y) &= 3500 + 2342 \\ \Rightarrow x + y + x - y &= 5842 \\ \Rightarrow 2x &= 5842 \end{aligned}$$ 4. **Divide both sides by 2 to solve for $x$:** $$\begin{aligned} 2x &= 5842 \\ \Rightarrow \cancel{2}x &= \cancel{2}2921 \\ x &= 2921 \end{aligned}$$ 5. **Interpretation:** Jarred has $2921$ contemporary DVD titles. **Final answer:** $\boxed{2921}$ contemporary titles.