91Ó°ÊÓ

When solving a linear equation, one of the essential steps is to isolate the variable. In this exercise, our goal is to solve for the variable \(y\). This means we need to get \(y\) by itself on one side of the equation. We start by looking at the equation:

$$\-2x + 3y = -9$$

Notice that \(y\) is currently being multiplied by 3 and there's an \(x\) term on the same side as \(y\). To isolate \(y\), we first need to move the \(-2x\) term to the other side of the equation.

We can do this by adding \(2x\) to both sides. Whatever we do to one side of an equation, we must do to the other side as well to keep it balanced.

So, we add \(2x\) to both sides:

$$-2x + 3y + 2x = -9 + 2x$$

This simplifies to:

$$3y = 2x - 9$$.

We've successfully moved the term with \(x\) to the other side, making it easier to focus on isolating \(y\)!

Linear equations are mathematical expressions that form a straight line when graphed on a coordinate plane. They usually have one or two variables and the highest power of the variables is one.

In our example, the original equation \(-2x + 3y = -9\) is a linear equation because:

• It has two variables, \(x\) and \(y\).
• Each variable is raised to the power of one.
• Its graph will be a straight line.

Working with linear equations is a foundational skill in algebra. These types of equations are straightforward once you understand the basic operations required to solve them, like addition, subtraction, multiplication, and division.

By transforming our equation to \(3y = 2x - 9\), we maintained its linear structure, making the next steps to isolate \(y\) clearer and more direct.

Algebraic manipulation involves using algebraic methods to simplify or solve equations. We use basic operations like adding, subtracting, multiplying, and dividing to transform equations into a simpler form.

In our case, we have simplified our equation to: $$3y = 2x - 9$$.

Now, we need to isolate \(y\) by dividing both sides of the equation by 3.

This step is crucial because it frees \(y\) from its coefficient (the number multiplying it).

So, we divide everything on both sides by 3: $$y = \frac{2x - 9}{3}$$.

This algebraic manipulation translates our equation into a form where \(y\) is isolated and expressed explicitly in terms of \(x\).

Remember, practice these steps often to become confident in solving linear equations. The more you practice algebraic manipulation, the easier it becomes to isolate variables and solve for them, no matter how complex the equations may seem initially.