\( \left.\qquad \begin{array}{l}\text { Solve the system of equations. } \\ \qquad \begin{array}{l}y=9 x-4 \\ y=6 x+8\end{array} \\ y=\square\end{array}\right) . \begin{array}{l}\frac{\square}{\square}\end{array} \)

To solve the system of equations \(y = 9x - 4\) and \(y = 6x + 8\), we can set the right-hand sides equal to each other because both equations equal \(y\): \[ 9x - 4 = 6x + 8 \] Now, let's solve for \(x\): 1. Subtract \(6x\) from both sides: \(9x - 6x - 4 = 8\) \(3x - 4 = 8\) 2. Add 4 to both sides: \(3x = 12\) 3. Divide by 3: \(x = 4\) Next, substitute \(x = 4\) back into either original equation to find \(y\). We can use the equation \(y = 9x - 4\): \[ y = 9(4) - 4 = 36 - 4 = 32 \] Thus, the solution to the system is \((x, y) = (4, 32)\) or \(y = 32\). Output: The history of solving systems dates back to ancient civilizations, where scholars in Mesopotamia developed methods to handle linear equations using geometric approaches. Fast forward to the 17th century, and you’d find René Descartes linking these ideas to his coordinate system, paving the way for algebra as we know it today! In real-world scenarios, solving systems of equations is crucial in fields such as economics for finding equilibrium prices, engineering for optimizing designs, and even ecology for modeling population dynamics. So the next time you encounter these equations, remember they hold the key to solving real-life problems!