91Ó°ÊÓ

One of the essential steps in solving linear equations is to isolate the variable of interest. In this exercise, we need to isolate the variable y. To start, look at the given equation:
\( x - 2y = -6 \). The goal is to get y by itself on one side of the equation.
To do this, we first subtract x from both sides to remove it from the left side of the equation. This results in:
\( -2y = -6 - x \).
After this step, we have successfully isolated the term containing y. Now, it becomes straightforward to solve for y as we'll see in the upcoming sections.

After isolating the variable term, the next step is to simplify the equation to solve for the variable. Here’s where we left off:
\( -2y = -6 - x \).
To get y by itself, we now divide each side of the equation by -2. This step looks like this:
\( y = \frac{-6 - x}{-2} \).
We still aren't done yet. To further simplify, we can break the fraction down into two separate fractions:
\( y = \frac{-6}{-2} + \frac{-x}{-2} \). Simplifying each fraction, we get:
\( y = 3 + \frac{x}{2} \). This form is much simpler and directly shows how y relates to x. Making equations as simple as possible helps to understand the relationship between variables better.

In many real-world applications, we encounter linear equations with two variables, such as the one in this exercise:
\( x - 2y = -6 \). These equations describe a line in a coordinate system, and solving them often means finding how one variable depends on the other.
When solving two-variable equations, focus on one variable first. Here, we chose y. By isolating y, we are able to express it in terms of x as seen in the simplified form:
\( y = 3 + \frac{x}{2} \).
Understanding this relationship is crucial for graphing the equation, predicting outcomes, or substituting y in another equation if you're dealing with a system of equations. Each step brings us closer to understanding the full picture of how the variables interrelate.