2) \( \left\{\begin{array}{l}x-y=2 \\ 2 x+3 y=9\end{array}\right. \)

To solve the system of equations, we can use substitution or elimination. Let's start with the first equation: \( x - y = 2 \). Rearranging it gives us \( x = y + 2 \). We can substitute this expression for \( x \) into the second equation \( 2x + 3y = 9 \). Substituting \( x \) gives us: \[ 2(y + 2) + 3y = 9 \] This simplifies to: \[ 2y + 4 + 3y = 9 \] Combine like terms: \[ 5y + 4 = 9 \] Subtract 4 from both sides: \[ 5y = 5 \] Dividing by 5 gives: \[ y = 1 \] Now substitute \( y = 1 \) back into the first equation to find \( x \): \[ x - 1 = 2 \] So: \[ x = 3 \] Thus, the solution to the system is \( x = 3 \) and \( y = 1 \). --- Historically, systems of equations have been used since ancient times, with early Egyptians utilizing methods to solve linear equations in a practical manner. The concept became more formalized with the work of mathematicians like René Descartes and Carl Friedrich Gauss, who developed systematic methods for handling such systems. In real-world situations, systems of equations can be seen in various fields such as economics, engineering, and physics. For instance, they are often used to calculate income and expenses in budget planning, where multiple constraints must be satisfied to achieve a desired financial outcome.