\( 5 x+10 y=80 \) and \( 3 x+4 y=42 \)

To understand the beauty of systems of equations, let's dive into their historical background! The roots of linear equations go back to ancient civilizations, where mathematicians like Babylonian scribes used geometrical methods to solve problems similar to our modern systems. Fast forward to the 17th century, and we have René Descartes who laid down the groundwork for algebra and geometry, bringing us closer to what we use today! Now, let's look at real-world applications! Systems of equations are everywhere, from planning production schedules in factories (where you need to balance inputs and outputs) to economics, where they help determine supply and demand. Engineers use these equations to design bridges or buildings, ensuring safety and stability by finding the optimal material distribution. So, next time you solve a system, remember you're tapping into a rich tapestry of human innovation!