1. The problem involves finding the length of diagonal AC in a quadrilateral where the diagonal is divided into two segments AE and EC. 2. Given: AC = 4x + 10, AE = 3x + y, EC = 2x + y. 3. Since AE and EC are parts of AC, their sum equals AC: $$AE + EC = AC$$ 4. Substitute the given expressions: $$3x + y + 2x + y = 4x + 10$$ 5. Combine like terms on the left side: $$3x + 2x + y + y = 4x + 10$$ $$5x + 2y = 4x + 10$$ 6. To isolate variables, subtract $4x$ from both sides: $$\cancel{5x} + 2y = \cancel{4x} + 10$$ $$x + 2y = 10$$ 7. This equation relates $x$ and $y$: $$x + 2y = 10$$ Without additional information, this is the simplified relation between $x$ and $y$ based on the diagonal lengths. Final answer: $$x + 2y = 10$$