9. Solve the system of equations using the method of your choice. \[ \begin{aligned} \frac{1}{3} x+y=1 \\ 2 x+3 y=12\end{aligned} \]

To solve the system of equations, let's first simplify the first equation: \[ \frac{1}{3} x + y = 1 \implies y = 1 - \frac{1}{3}x. \] Now, substitute \(y\) into the second equation: \[ 2x + 3\left(1 - \frac{1}{3}x\right) = 12. \] This simplifies to: \[ 2x + 3 - x = 12 \implies x + 3 = 12 \implies x = 9. \] Now, substituting \(x = 9\) back into the equation for \(y\): \[ y = 1 - \frac{1}{3}(9) = 1 - 3 = -2. \] So, the solution to the system of equations is \(x = 9\) and \(y = -2\). Let’s summarize: The solution is \( (9, -2) \). Now let’s think about how this might play out in real life. Say you're managing a budget where the first equation represents the balance of your spending, while the second outlines total expenses. Understanding how to solve the equations allows you to effectively allocate funds, assuring you always stay within your limits. Also, envision a scenario where you're attempting to optimize resources for a project. By making use of systems of equations, you can mathematically model relationships between resources and objectives, ensuring you utilize your assets in the most efficient way possible. This mathematical skill is like a superpower for real-world planning and problem-solving!