1) \( \begin{aligned} 3(x-4)= & 2(-2 x+1) \\ 3 x-(1) & =4 x+(2) \\ +4 x & +4 x\end{aligned} \)

First off, let's solve the equation step by step! Starting with the first equation \( 3(x - 4) = 2(-2x + 1) \): 1. Distribute: - Left side: \( 3x - 12 \) - Right side: \( -4x + 2 \) So, we have: \[ 3x - 12 = -4x + 2 \] 2. Add \( 4x \) to both sides: \[ 3x + 4x - 12 = 2 \] \[ 7x - 12 = 2 \] 3. Add 12 to both sides: \[ 7x = 14 \] 4. Divide by 7: \[ x = 2 \] Now, for the second equation, \( 3x - 1 = 4x + 2 \): 1. Subtract \( 3x \) from both sides: \[ -1 = x + 2 \] 2. Subtract 2 from both sides: \[ -1 - 2 = x \] \[ x = -3 \] Now you have two solutions: \( x = 2 \) from the first equation and \( x = -3 \) from the second! Each equation gives you a different solution. Splitting up can sometimes lead to some interesting surprises! Now, as for useful tips and insights: When solving linear equations, always remember to keep your operations balanced: whatever you do to one side, do to the other! Also, double-check your distributions and combine like terms carefully; errors often sneak in here!