\( x+y=2 \) and \( x^{2}+y^{2}=10 \)

Let's break this down in a fun way! Did you know that you can create a system of equations that seems simple but opens doors to various fascinating mathematical insights? This problem puts us on a mathematical adventure where we can explore the unknowns, \(x\) and \(y\), like detectives solving a mystery! Now, for a real-world application of this kind of mathematical problem, think of it like budgeting your expenses! Say \(x\) represents your spending on entertainment and \(y\) on groceries. The first equation tells you that you want to keep your total spending to 2 units, while the second one gives you a slightly complicated constraint on how your expenditures relate. By working through such equations, you can find the optimal way to manage your budget without overspending!