7) \( 3 a x-3 a y-x+y \)

To simplify the expression \( 3 a x - 3 a y - x + y \), we can first rearrange it for clarity: \( (3 a x - x) + (-3 a y + y) \). Now, factor out common terms from each group. From the first group, we can factor out \( x \), giving us \( x(3 a - 1) \). From the second group, we can factor out \( y \), resulting in \( -y(3 a - 1) \). Thus, we can combine the two parts to get: \((3 a - 1)(x - y)\). This expression reminds us of how factoring can help us express complex algebraic forms in a cleaner way, making it easier to analyze or solve for specific values of \( a, x, \) and \( y \). When dealing with algebraic expressions, a common mistake is forgetting to distribute negative signs correctly. For instance, in the original expression, ensuring that you distribute the subtraction properly, especially if you're rearranging or combining terms, is crucial. Always double-check your signs before moving forward!