1) \( \left\{\begin{array}{l}x+2 y=4 \\ -2 x+5 y=10\end{array}\right. \)

To solve this system of equations, let's use the substitution or elimination method. If we take the first equation, \( x + 2y = 4 \), you can express \( x \) in terms of \( y \) as \( x = 4 - 2y \). Now, substitute this into the second equation: \(-2(4 - 2y) + 5y = 10\). Simplifying, we get: \(-8 + 4y + 5y = 10\) or \(9y = 18\). Therefore, \(y = 2\). Substituting \(y\) back into \(x = 4 - 2(2)\) gives \(x = 0\). So the solution is \( (0, 2) \). But wait—do you realize that systems like this are used in real-life situations? They can represent anything from budgeting to predicting outcomes in business scenarios. Think of it as a mathematical way of balancing competing needs; a savvy move that can mean the difference between profitability and loss! Exploring further, if you're into challenging puzzles, systems of equations can be a foundation for deeper adventures in math! Delve into matrix operations or even linear programming! Get those cognitive gears turning while you tackle real-world challenges with creatively structured equations. Math can be more than just numbers; it's your mental playground!