\( \begin{array}{l}\text { a. } 2 y-x=5 \\ -4 y+2 x=-10\end{array} \) \( \square \) A) one solution \( \square \) B) infinitely many solutions \( \square \) C) no solution \( \square \) D) consistent \( \square \) E) inconsistent independent \( \square \) G) dependent

To better understand the system of equations given, let's first dive into the historical context of linear equations. These types of equations have been studied for centuries, dating back to ancient civilizations like Babylon where early mathematicians used geometric methods. The development of algebra in the Islamic Golden Age further advanced this field, leading to the systematic solving of linear equations. Now, on to real-world applications! Imagine you're balancing financial accounts or inventory in a business. Systems of linear equations can represent these scenarios, helping you to determine things like the optimal levels of production or budgeting. This mathematical tool is invaluable in industries from economics to engineering, as it simplifies complex relationships into manageable calculations.