2. \( \begin{aligned} 3 m+4 n & =8 \\ 5 m-6 n & =7\end{aligned} \)

To solve the system of equations, you can use methods like substitution or elimination. For instance, if you choose elimination, you can first multiply the first equation by 5 and the second by 3, so that the coefficients of \( m \) line up. You would get \( 15m + 20n = 40 \) and \( 15m - 18n = 21 \). Subtracting these two equations cancels out \( m \), allowing you to solve for \( n \). Once you find the value of \( n \), you can substitute it back into one of the original equations to find \( m \). This process highlights the beauty of algebra in solving complicated problems systematically! Alternatively, a quick mental check of your solutions or using online tools like graphing calculators can save time and reduce errors in complex calculations. Just remember, it's easy to make simple mistakes in sign or arithmetic, so double-checking each step is key to ensuring a smooth journey to the correct answer!