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How To Really Solve This Tricky Algebra Problem? (IV)

Difficulty alert: Math Olympiad level problem!

How To Really Solve This Tricky Algebra Problem? (IV) — An image showing the following equations on the left: x² + y² = 7 and x³ + y³ = 10. On the right, the following equation is written: x + y = ??. Below this equation, the following information is provided in a box: x,y ∈ ℝ.
Image created by the author

Welcome to the fourth entry in the tricky algebra problem series. This problem features a system of equations involving two unknowns: x and y, both of which are limited to the set of all real numbers. To start, you are provided with the following two equations:

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