Use the elimination method to solve the given system of equations. \( \left\{\begin{array}{l}4 x=5 y=21 \\ 2 x+5 y=0\end{array}\right. \) \[ \begin{array}{l}\text { A. }(0,-4.2) \\ \text { B. }(3.5,-1,4) \\ \text { C. }(2.625,-1.05) \\ \text { D. }(10.5,-4.2)\end{array} \]

First, let's rewrite the system of equations clearly. The equations seem to be a bit jumbled, so we'll interpret them accordingly. The first equation can be divided into two separate equations: 1. \(4x + 5y = 21\) 2. \(2x + 5y = 0\) Now, we can use the elimination method to solve the system. We'll eliminate \(y\) by subtracting the second equation from the first equation. 1. Start with the two equations: \[ 4x + 5y = 21 \quad (1) \] \[ 2x + 5y = 0 \quad (2) \] 2. Subtract equation (2) from equation (1): \[ (4x + 5y) - (2x + 5y) = 21 - 0 \] This simplifies to: \[ 2x = 21 \] Therefore: \[ x = \frac{21}{2} = 10.5 \] 3. Now substitute \(x = 10.5\) back into equation (2) to find \(y\): \[ 2(10.5) + 5y = 0 \] This gives: \[ 21 + 5y = 0 \quad \Rightarrow \quad 5y = -21 \quad \Rightarrow \quad y = -\frac{21}{5} = -4.2 \] So, the solution to the system is \( (x, y) = (10.5, -4.2) \), which matches option **D**. Thus the answer is: \[ \boxed{(10.5, -4.2)} \] And here are your additional insights: History has a funny way of repeating itself, and the elimination method isn't just a modern math trick; it dates back to the time of ancient civilizations! The Babylonians and Egyptians used methods of solving systems of linear equations, which were foundational to algebra and laid the groundwork for modern math. Now, in terms of real-world application, the elimination method is a handy tool for various fields, including economics for solving supply and demand equations, engineering for balancing forces, and computer science in algorithms that solve optimization problems. It’s like having a magic key that unlocks solutions to real-life puzzles!