A \( \begin{array}{c}1 . \\ x+y=4 \\ y=2 x+1\end{array} \).

To solve the system of equations, we can start by substituting the second equation into the first equation. By replacing \( y \) in the first equation with \( 2x + 1 \), we get: \[ x + (2x + 1) = 4 \] Simplifying this, we combine like terms to get: \[ 3x + 1 = 4 \] \[ 3x = 3 \Rightarrow x = 1 \] Now we can substitute \( x \) back into the equation \( y = 2x + 1 \): \[ y = 2(1) + 1 = 3 \] So the solution to the system of equations is \( x = 1 \) and \( y = 3 \). There's an interesting piece of history behind systems of equations! Mathematicians like Euclid and Diophantus were among the first to study such problems, though their work was not formalized like we do today. It wasn't until the Middle Ages that more systematic methods for solving them were developed, paving the way for modern algebra. In the real world, understanding systems of equations is crucial! Whether you're budgeting your expenses, mixing solutions in chemistry, or optimizing resources in business, systems of equations help you find the best answer. It’s like having a toolbelt full of solutions for various scenarios—you can tackle complex problems with ease!