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In precalculus, a function is defined as a special relationship between values, where each input (commonly denoted as x) has a single output (often denoted as y). To define functions algebraically, equations are used to clearly describe the input to output relationship. The equation given in the original exercise, \(xy+2y=1\), poses the question of whether it defines y as a function of x.

The process of determining this involves a few algebraic manipulations to express y solely in terms of x. If this can be done without ambiguity—that is, if for each x there is one and only one y—an equation defines y as a function of x. The provided step-by-step solution successfully demonstrates that y can be isolated, which leads us to conclude that y is indeed a function of x.

Isolating variables is a fundamental algebraic skill that's especially important when defining functions or solving equations. To effectively isolate a variable means to get the variable alone on one side of the equation, with a coefficient of 1. In the step-by-step solution, isolating the variable y was the initial step. By moving all terms involving y to one side and all other terms to the opposite side, you simplify the process of defining the function.

In the example equation, the term \(xy\) was subtracted from both sides leading to \(2y = 1 - xy\). Isolating variables can sometimes require additional steps like dividing by coefficients or factoring, but the goal always remains the same: one variable alone on one side to define its value in terms of the other variables.

Factoring is an essential technique in algebra that simplifies expressions and solves equations. It involves writing an expression as a product of its factors. This can be particularly useful when dealing with polynomial expressions, but it is also applicable to equations where the variable needs to be isolated, as was necessary in the example problem.

After subtracting \(xy\) from both sides, the equation becomes \(2y = 1 - xy\). To isolate y, the common factor of y was factored out from the terms on the left-hand side, resulting in \(y(2 + x) = 1\). From this form, we see that each term involving y is part of a product, which can be easily managed to solve for y. Factoring not only simplifies expressions but also reveals underlying structures that aid in finding solutions to algebraic problems.