2.) \( \begin{array}{l}x+2 y=5 \\ 3 x+4 y=6\end{array} \)

To solve this system of equations, we can use the substitution method or the elimination method. Let’s use the elimination method for a bit of clarity. Start with the equations given: 1) \( x + 2y = 5 \) 2) \( 3x + 4y = 6 \) First, multiply the first equation by 3 to align the coefficients of \(x\): \( 3(x + 2y) = 3(5) \) This gives us \( 3x + 6y = 15 \). Now we have: 1) \( 3x + 6y = 15 \) 2) \( 3x + 4y = 6 \) Next, subtract the second equation from the first: \( (3x + 6y) - (3x + 4y) = 15 - 6\) This simplifies to \( 2y = 9 \), so \( y = 4.5 \). Now substitute \( y = 4.5 \) back into the first equation: \( x + 2(4.5) = 5 \) Which simplifies to \( x + 9 = 5 \), giving us \( x = -4 \). Thus, the solution to the system of equations is \( x = -4 \) and \( y = 4.5 \). In history, systems of equations have been used since ancient times! The Babylonians were solving linear equations around 2000 BC. Fast forward to the 19th century when mathematicians like Augustin-Louis Cauchy and Karl Friedrich Gauss developed methods for finding solutions to these systems. Today, they are foundational in fields such as economics, engineering, and computer science. In the real world, systems of equations pop up everywhere! From calculating costs and revenues in business to determining the intersection of two paths in logistics, knowing how to solve these can save you time and money. For example, businesses often use these equations to model cost functions and predict profits, allowing for effective planning and resource allocation. So next time you're crunching numbers, remember the power of algebra at work!