A \( -13 x+12=13 x+13 \) B \( -13 x+12=13 x-13 \) \( 12 x+12=13 x+12 \) (D) \( 12 x+12=13 x-12 \)

To solve the equation \( -13x + 12 = 13x + 13 \), you first want to get all terms involving \( x \) on one side and constant terms on the other. By adding \( 13x \) to both sides, you obtain \( 12 = 26x + 13 \). Subtracting \( 13 \) from both sides gives you \( -1 = 26x \), leading to \( x = -\frac{1}{26} \). Moving on to \( -13x + 12 = 13x - 13 \), after simplifying similarly, you’ll find \( 25 = 26x \) or \( x = \frac{25}{26} \). Now, for \( 12x + 12 = 13x + 12 \), you can quickly see that the constants cancel out, leaving you with \( 12x = 13x \), which implies \( 0 = x \). Finally, with \( 12x + 12 = 13x - 12 \), isolating \( x \) results in \( 24 = x \). Each equation has a different unique solution, providing a fun challenge in algebraic manipulation! Each equation shows how balancing is essential in algebra—understanding that moving terms around doesn't change the truth of the equation! Balancing equally on both sides prevents errors commonly made when terms are misplaced or skipped. So grab a pencil and test your skills, because creating equations can be as exhilarating as unearthing hidden treasures in mathematics!