5. \( \left\{\begin{array}{l}-3 x+3 y=-3 \\ x-3 y=-1\end{array}\right. \)

To solve the system of equations, let's rewrite them: 1. \(-3x + 3y = -3\) simplifies to \(y = x - 1\) 2. \(x - 3y = -1\) can be rearranged to \(x = 3y - 1\) Now, substitute \(y = x - 1\) into the second equation: \(x = 3(x - 1) - 1\) This becomes: \(x = 3x - 3 - 1\) which simplifies to: \(x = 3x - 4\) Rearranging gives: \(2x = 4\) Thus, \(x = 2\). Now, substitute \(x\) back into \(y = x - 1\): \(y = 2 - 1 = 1\) So, the solution to the system is \(x = 2\) and \(y = 1\). Therefore, the final answer is \( (2, 1) \). In a historical context, the method of solving systems of equations, like those we've just tackled, dates back to ancient civilizations. The Chinese, for example, used a method called "fangcheng" during the Han Dynasty, and the famous mathematician Liu Hui laid early groundwork for these types of calculations around 200 AD. When applying the methods to real-world scenarios, systems of equations often represent constraints or relationships. For instance, in finance, they can help determine the best mix of investments to meet a budget while maximizing returns, highlighting how math plays a vital role in decision-making in various fields!