Final answer:
When you multiply both sides of an equation by a common multiple of the denominators that is not the LCM, you still maintain equality but may introduce extra factors that require further simplification.
Step-by-step explanation:
When solving equations that involve fractions, one common step is to eliminate the denominators by multiplying both sides of the equation by a common multiple of those denominators. Typically, the least common multiple (LCM) is used because it simplifies the calculations. However, if a multiple that is not the LCM is used, the equation will not necessarily be incorrect, but it may be more complex and require additional simplification. When you multiply both sides by a common multiple of the denominators that is not the LCM, you do not change the equality of the equation, but you may introduce additional factors that are not common to every term, potentially complicating the equation further. It is important to ensure that any multiplication of an equation is done to every term on both sides.
Example: Multiplying the equation ½ + ¼ = x by 4, which is not the LCM but a multiple of the denominators, yields (4*½) + (4*¼) = 4*x, which simplifies to 2 + 1 = 4x. Had we used the LCM (which is 4), the result would be the same. However, if we chose a multiple like 8, the simplification would involve an additional step: (8*½) + (8*¼) = 8x, simplifying to 4 + 2 = 8x and then reducing to 1 + ½ = 2x.