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Understanding the concept of functions is fundamental in algebra. Basically, a function associates each element in a given domain with exactly one element in the range. In simpler terms, if you input a certain number into the function, you get a single result out. To determine if a given equation represents a function of a variable, say, \(y\) as a function of \(x\), you look for one-to-one correspondence between \(x\) and \(y\). In the example \(2x+3y=4\), we can manipulate the equation to isolate \(y\). Once \(y\) is by itself on one side of the equation, we can analyze the relationship. If for every value of \(x\), there is only one possible \(y\), then \(y\) is indeed a function of \(x\). As shown in the solution, after rearranging and simplifying, each \(x\) value corresponds to exactly one \(y\) value, satisfying the criteria for a function.

Linear equations are the simplest kind of algebraic function and they play a huge role in various fields like economics, physics, and engineering. A linear equation looks to involve terms of the variable to the first power (like \(x\) or \(y\), but not \(x^2\), for example). Here's how to solve them:

• Rearrange the terms to get the unknown variable on one side of the equation.
• This often involves moving terms around by adding, subtracting, multiplying, or dividing both sides by the same number to maintain the balance of the equation.
• In our exercise, once we had \(3y = 4 - 2x\), we divided everything by 3 to solve for \(y\).

By following these steps, we found that \(y = \frac{4 - 2x}{3}\), which means we have isolated \(y\) and therefore solved the linear equation. An important aspect here is keeping the equation balanced - whatever you do to one side, you must also do to the other.

Algebraic manipulation is the process of transforming mathematical expressions into different equivalent forms, making them easier to understand or solve. The process includes various techniques:

• Addition or subtraction of the same number or expression from both sides of an equation (As we did by subtracting \(2x\) from both sides).
• Multiplication or division of both sides of an equation by the same non-zero number (For instance, dividing by 3 to isolate \(y\)).
• Expanding expressions using the distributive property (if applicable).
• Factoring expressions, which is the reverse process of expanding (useful when dealing with quadratic equations or higher).

These manipulations help in solving equations, simplifying expressions, and in the case of our problem, isolating a variable to determine if it represents a function of another variable. Such skills are quintessential for solving a wide range of problems in algebra.