Simplify the following expression completely: \[ \frac{2 x-6 y}{4 x-12 y}, \quad x \neq 3 y \] Answer: The simplified fraction is: \[ \quad x \neq 3 y \]

To simplify the expression \(\frac{2x - 6y}{4x - 12y}\), we begin by factoring both the numerator and the denominator. The numerator can be factored as \(2(x - 3y)\) and the denominator as \(4(x - 3y)\). This gives us: \[ \frac{2(x - 3y)}{4(x - 3y)} \] Now, we can cancel out the common factor \(x - 3y\) (keeping in mind that \(x \neq 3y\)): \[ \frac{2}{4} = \frac{1}{2} \] Thus, the simplified fraction is: \[ \frac{1}{2} \quad (x \neq 3y) \] When working through algebraic simplifications, a common mistake is neglecting to consider restrictions on the variable involved. Here, since we cancelled \(x - 3y\), we've introduced a caveat that \(x\) must not equal \(3y\). Ignoring such conditions can lead to undefined expressions. For further understanding of fractions, you might want to check out resources on rational expressions and their simplification methods. Books focusing on algebra often have practice problems that can sharpen your skills in factoring and reducing expressions effectively!