I do apologize. Let me clarify; I was not saying that anything you have been taught is wrong. There frequently are different but equally valid ways to solve a problem.
What I am saying is that ”clearing fractions,” that is “eliminating all fractions from the equation,“ is USUALLY the least error prone method when it can be applied without great difficulty.
There are, moreover, two methods of clearing fractions. One is to multiply both sides of the equation by the least common denominator of all the denominators in the equation. The other is to multiply both sides of the equation by the product of all the denominators. Both methods are equally valid.
Common denominator method.
[MATH]\dfrac{1}{2} * (y - 5) = \dfrac{1}{4} * (y - 1).[/MATH]
It is obvious that 4 is the least common multiple of the denominators.
[MATH]4 * \dfrac{1}{2} * (y - 5) = 4 * \dfrac{1}{4} * (y - 1) \implies 2(y - 5) = (y - 1) \implies[/MATH]
[MATH]2y - 10 = y - 1 \implies 2y - y = 10 - 1 \implies y = 9.[/MATH]
However, finding a least common multiple of the denominators is often a lot of work, and that work is never necessary. You can always clear fractions by multiplying by the product of the denominators. In this case 2 * 4 = 8.
[MATH]\dfrac{1}{2} * (y - 5) = \dfrac{1}{4} * (y - 1) \implies 8 * \dfrac{1}{2} * (y - 5)= 8 * \dfrac{1}{4} * (y - 1) \implies [/MATH]
[MATH]4(y - 5) = 2(y - 1) \implies 4y - 20 = 2y - 2 \implies 4y - 2y = 20 - 2 \implies 2y = 18 \implies y = 9.[/MATH]
You NEVER have to use a least common multiple of the denominators to clear fractions although it is sometimes the fastest way to proceed.