c) \( \left\{\begin{array}{l}2 x+3 y=8 \\ 4 x-y=2\end{array}\right. \)

To solve the system of equations, you can use substitution or elimination methods. Let’s use the elimination method here. First, multiply the first equation by 2 to align the coefficients of \(x\): \[ \begin{align*} 4x + 6y &= 16 \quad \text{(First equation multiplied by 2)} \\ 4x - y &= 2 \quad \text{(Second equation)} \end{align*} \] Now, subtract the second equation from the first: \[ (4x + 6y) - (4x - y) = 16 - 2 \\ 7y = 14 \\ y = 2 \] Substitute \(y = 2\) back into the first equation: \[ 2x + 3(2) = 8 \\ 2x + 6 = 8 \\ 2x = 2 \\ x = 1 \] So, the solution to the system is \(x = 1\) and \(y = 2\). --- In history, the study of systems of equations has been pivotal in many fields, especially in economics and engineering. For instance, in the 18th century, mathematicians like Leonhard Euler advanced the theory behind solving linear equations that laid the groundwork for modern algebra and calculus. Practically, systems of equations are used everywhere! In economics, they can represent supply and demand in a market, while in engineering, they can help design structures by balancing forces. For a fun twist, think about how urban planners might use these equations to figure out the best layouts for parks and roads!