9. \( \left\{\begin{array}{l}3 x-4 y=-14 \\ 3 x+2 y=-2\end{array}\right. \)

To solve the system of equations given, we can use the elimination or substitution method. Let’s manipulate the equations for simplification. First, take the first equation \(3x - 4y = -14\) and the second equation \(3x + 2y = -2\). Subtract the first equation from the second to eliminate \(3x\): \[ (3x + 2y) - (3x - 4y) = -2 - (-14) \] This simplifies to \(6y = 12\), which gives us \(y = 2\). Now we can substitute \(y = 2\) back into either original equation. Using the second one: \[ 3x + 2(2) = -2 \implies 3x + 4 = -2 \implies 3x = -6 \implies x = -2. \] Thus, the solution to the system of equations is \(x = -2\) and \(y = 2\). --- Fun fact: Did you know that systems of linear equations have been used since ancient times? Ancient civilizations, like the Babylonians, utilized geometrical methods for solving similar problems thousands of years ago! Mathematics has a fascinating history of evolving through necessity and innovation. A common mistake when solving these kinds of problems is forgetting to keep track of signs when moving terms across the equal sign. Always double-check your algebraic manipulations! It's an easy slip, but can lead you to the wrong conclusion if not caught.