91Ó°ÊÓ

Understanding how to isolate variables is crucial in mathematics, particularly when you're trying to find the value of one variable in terms of another. In the equation \(2x + 5y = 10\), you need to obtain \(y\) as a standalone subject. This is often the first step in solving equations and can be accomplished by performing the same operation on both sides of the equation, maintaining the balance.

We start by getting rid of the term that does not contain our variable of interest. In this case, we want to isolate \(y\), so we remove the \(2x\) from the left side by subtracting \(2x\) from both sides. This leaves us with \(5y = -2x + 10\). Once \(y\) is by itself on one side of the equation, it's easier to grasp its relationship with \(x\), which is essential for determining if a function is present.

Once you have isolated \(y\), the next stage is solving the equation to express \(y\) solely in terms of \(x\). This means you want a formula where you plug in an \(x\) value and get a corresponding \(y\) value. With our isolated equation \(5y = -2x + 10\), we proceed by dividing every term by the coefficient of \(y\), which is 5 in this case.

The outcome of this operation is \(y = -\frac{2}{5}x + 2\), which reveals a clear relationship between \(x\) and \(y\). Dividing by the coefficient ensures that \(y\) stands alone, and we can clearly see that for each \(x\) value there will be one unique \(y\) value. This step is fundamental in algebra and enables a consistent method for finding solutions to variable problems.

A function is essentially a relationship that assigns exactly one output to each input. In other words, if you have an input variable (often \(x\)) and an output variable (often \(y\)), and each input \(x\) value pairs with exactly one output \(y\) value, then \(y\) is a 'function' of \(x\).

From our work in previous steps, we arrived at the equation \(y = -\frac{2}{5}x + 2\), which indeed outlines \(y\) as a function of \(x\). Here, no matter what value we substitute for \(x\), we will only get one value for \(y\). It’s a one-to-one relationship that’s fundamental to understanding functions in mathematics. This concept enables mathematicians and scientists to model and predict behavior in a myriad of applications, from physics to economics.