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In algebraic equations, isolating the variable means rearranging the equation so that one side contains the variable of interest, and the other side contains the other terms. This is a fundamental step to solve equations as it helps to focus on the variable and find its value. For our problem, the variable we need to isolate is part of the term \(3y\) in the equation \(-x + 3y = 9\). To isolate \(y\), we start by removing the term that does not involve \(y\) from the left side of the equation.

• Identify the term involving the variable you want to isolate. In this case, it's \(3y\).
• Eliminate other terms from the same side of the equation, here by adding \(x\) to both sides, resulting in \(3y = x + 9\).

These actions help ensure that the equation is balanced and move steps closer to isolating \(y\). By dealing with one term at a time, complex equations become simpler.

Once the variable has been isolated, the next task focuses specifically on obtaining a direct expression for that variable. For equations like \(3y = x + 9\), the main goal is to express \(y\) in terms of other known quantities. Here is how this works:

- Recognize that \(3y\) means 3 times \(y\). To solve for \(y\), you then need to divide both sides by 3.- This results in the equation \(y = \frac{x + 9}{3}\).

Why is this division important? It's because you've now expressed \(y\) explicitly, meaning you've got \(y\) all by itself on one side of the equation. This means you solve for whatever \(x\) might be, you've got the answer for \(y\). By the end of this process, you've unveiled a clearer picture of how \(y\) changes with \(x\).

Linear equations are fundamental in algebra. They involve relationships that can be graphed as straight lines on a Cartesian plane. These equations are often written in the form \(Ax + By = C\).

- **Concept**: The term 'linear' indicates a straight line when plotted. The equation \(-x + 3y = 9\) is a linear equation. The variables appear only to the first power and their graph is a straight line.- **Slope-Intercept Form**: The final form of \(y = \frac{x + 9}{3}\) resembles the slope-intercept form \(y = mx + b\), where 'm' represents the slope and 'b' the y-intercept. In our context, the slope \(m\) is \(\frac{1}{3}\) and the y-intercept \(b\) is 3.

Linear equations are beneficial for modeling simple relationships where change occurs at a constant rate. Understanding this can help in predicting trends and making calculations in various fields like physics and economics.