91Ó°ÊÓ

Understanding how to isolate a variable is crucial in solving equations. In the provided exercise, the goal was to solve for the variable \(y\). To do this, we first need to isolate the term that contains \(y\). This means getting \(y\) by itself on one side of the equation. For our equation, \(-3x + 2y = 5\), we start by moving any terms that do not contain \(y\) to the other side of the equation.

Here’s the process step-by-step:
- Add \(3x\) to both sides: \(-3x + 2y + 3x = 5 + 3x \).
- This simplifies to: \ 2y = 3x + 5 \.
Now, the term containing \(y\) is isolate. It’s easier to solve for \(y\) when the expression is in this form. The key here is to ensure that every step is balanced. This means performing the exact same operation on both sides of the equation.

Algebraic manipulation is the process of using mathematical operations to simplify and solve equations. This can involve addition, subtraction, multiplication, and division, as well as distributing and factoring terms. When isolating a variable like we did in the exercise, algebraic manipulation helps move terms around while keeping the equality of the equation.

Let's revisit our equation after the first step: \(2y = 3x + 5\). To fully isolate \(y\), we need to perform algebraic manipulation on this equation. We do this by dividing every term by 2:
- Divide both sides by 2: \ \frac{2y}{2} = \frac{3x + 5}{2} \.
- Simplify: \ y = \frac{3x + 5}{2} \.
By dividing all terms, we keep the equality balanced while solving for \(y\). This methodical manipulation brings us closer to finding the solution.

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations graph as straight lines and take the general form of \(ax + by = c\). The given exercise starts with the linear equation \ -3x + 2y = 5 \. Our goal is to solve this equation for \(y\).

Steps to solve a linear equation typically include:

• Identifying the variable that you need to solve for.
• Using algebraic manipulation to isolate this variable.
• Balancing both sides throughout every step to maintain equality.

After we isolate \(2y\), our target term, we divided all terms by 2 to find the final solution. Understanding these principles helps in solving more complex equations. Being proficient with linear equations is fundamental in algebra and many real-world applications.