91Ó°ÊÓ

Isolating variables is a fundamental skill in algebra. It refers to the process of getting a specific variable alone on one side of an equation. This often helps in simplifying complex equations and understanding the relationship between different variables.

In the given exercise, we started with the equation \(\frac{1}{2}x - \frac{1}{3}y = 2\). Our goal was to express \(y\) as a function of \(x\).

The first step involved isolating the term containing \(y\). We did this by subtracting \(\frac{1}{2}x\) from both sides: \(-\frac{1}{3} y = 2 - \frac{1}{2} x\).

Next, we eliminated the fraction by multiplying both sides by -3. This step was crucial because it simplified the equation and allowed us to clearly solve for \(y\). By performing these operations, we made \(y\) the subject of the formula, achieving isolation.

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. Linear equations form straight lines when graphed, hence the name 'linear.'

The given equation \(\frac{1}{2} x - \frac{1}{3} y = 2\) is a perfect example of a linear equation. It includes two variables, \(x\) and \(y\), and can be rewritten in a form that shows \ y\ as a linear function of \(x\).

After isolating \(y\) and simplifying, we obtained the equation \(y = -6 + \frac{3}{2} x\). This is another representation of a linear equation, showing a clear linear relationship between \(x\) and \(y\).

A function of \(x\) (\(y = f(x)\)) is an expression where \(y\) is defined in terms of \(x\). This means that for every unique input of \(x\), there's a corresponding output of \(y\).

In the exercise, we started with a linear equation and transformed it into a function of \(x\). To do this, we arranged the equation to solve for \(y\), giving us \(y = -6 + \frac{3}{2} x\).

This result tells us how \(y\) changes with respect to \(x\). For example, if \(x\) increases, the term \ \frac{3}{2} x \ will increase, thus increasing \(y\).

So, the transformation of the given equation into \(y = -6 + \frac{3}{2} x\) makes it much easier to see how \(y\) behaves as a function of \(x\). This is a core principle in algebra, helping students understand the dependencies between variables.