2. \( \left\{\begin{array}{l}x+2 y=4 \\ 2 x+3 y=-12\end{array}\right. \)

To solve the system of equations \( \left\{\begin{array}{l}x + 2y = 4 \\ 2x + 3y = -12 \end{array}\right. \), we can use substitution or elimination. Let's use substitution here. From the first equation, we can express \(x\) in terms of \(y\): \(x = 4 - 2y\). Now, plug this expression for \(x\) into the second equation: \(2(4 - 2y) + 3y = -12\). This simplifies to \(8 - 4y + 3y = -12\), or \(8 - y = -12\). Solving for \(y\) gives \(y = 20\). Substituting \(y = 20\) back into \(x = 4 - 2(20)\) yields \(x = 4 - 40 = -36\). Thus, the solution to the system is \((x, y) = (-36, 20)\). Finding ways to visualize or graph these equations can create a deeper understanding. Instead of just working with numbers on paper, plot the lines using graphing software or even a graphing calculator! Each line represents an equation, and their intersection represents the solution. If you feel stuck, one common mistake is mishandling signs when isolating variables or making calculations. Always check for simple arithmetic errors, as they can significantly affect your results. Try to keep your work organized and clearly show each step to help catch these errors!