1. **State the problem:** Ella sells small prints for 50 each and large prints for 75 each. She sells a total of 52 prints and earns 2975 in total. We need to find how many small and large prints she sold. 2. **Define variables:** Let $x$ be the number of small prints and $y$ be the number of large prints. 3. **Write the system of equations:** $$\begin{cases} x + y = 52 \\ 50x + 75y = 2975 \end{cases}$$ 4. **Solve the first equation for $y$:** $$y = 52 - x$$ 5. **Substitute $y$ into the second equation:** $$50x + 75(52 - x) = 2975$$ 6. **Distribute 75:** $$50x + 3900 - 75x = 2975$$ 7. **Combine like terms:** $$-25x + 3900 = 2975$$ 8. **Subtract 3900 from both sides:** $$-25x + \cancel{3900} - \cancel{3900} = 2975 - 3900$$ $$-25x = -925$$ 9. **Divide both sides by -25:** $$\frac{-25x}{\cancel{-25}} = \frac{-925}{\cancel{-25}}$$ $$x = 37$$ 10. **Find $y$ using $y = 52 - x$:** $$y = 52 - 37 = 15$$ **Final answer:** Ella sold 37 small prints and 15 large prints.