\( \begin{array}{lr}\begin{array}{lr}2 X+3 Y=10 & \text { ii) } \\ 3 X-2 Y=7 \\ 4 X-Y=5 & 5 X+4 Y=1 \\ \text { iii) } X+Y=7 & \text { iv) } 4 X+5 Y=12 \\ 2 X-4 Y=1 & 2 X-3 Y=-5 \\ 3 X+2 Y=10 & \text { vi) } 10 X+2 Y=6 \\ 2 X-4 Y=6 & 2 X+2 \frac{3}{7} Y=6 \\ \text { v) } & \\ \text { viii) } X+3 Y=8 \\ 2 \frac{5}{8}+2 \frac{3}{5} Y=5 & 7 X+5 Y=10\end{array} \\ 3 \frac{6}{8}+2 \frac{6}{7} Y=10 & \end{array} \)

Ever wondered about the origins of systems of linear equations? They date back to ancient civilizations like Egypt and Babylon, where they were used for practical day-to-day calculations, such as crop distributions and architectural designs. Fast forward a few thousand years, and mathematicians like René Descartes and Carl Friedrich Gauss laid the groundwork for modern algebraic methods, making it possible to solve these equations with flair! In real-world scenarios, linear equations come into play all the time! For instance, businesses use them to determine profit margins, while engineers use them to calculate loads and forces in structures. By modeling relationships and constraints, these equations help keep everything from finances to construction projects on track. So, next time you encounter a set of equations, remember they’re not just abstract numbers—they’re the keys to solving real-life puzzles!