91Ó°ÊÓ

The key to solving any algebraic equation is to isolate the variable you want to solve for, in this case, \(y\). To isolate a variable means to get the variable on one side of the equation by itself. Start by identifying the terms that include the variable. The provided equation is: \[5x + 2y = 9x - 3y\]
Our goal is to solve for \(y\).

Follow these steps to isolate \(y\):

• Move all terms that include \(y\) to one side of the equation.
• Use algebraic operations, such as addition, subtraction, multiplication, or division to isolate \(y\).

This step-by-step method can be applied to any algebraic equation where you need to isolate a particular variable.

Algebraic manipulation involves rearranging equations using various algebraic operations. In the given equation, our goal is to solve for \(y\) by manipulating the terms. Here's how to do it:

1. First, identify the terms involving \(y\) and move them to one side:
\[5x + 2y + 3y = 9x\]

2. Combine like terms to simplify: \[5x + 5y = 9x\]

3. Next, move the \(x\) terms to the opposite side so that all \(y\) terms are on one side:
\[5y = 9x - 5x\]

4. Simplify the equation by combining the \(x\) terms:
\[5y = 4x\]

5. Finally, solve for \(y\) by dividing both sides by 5:
\[y = \frac{4x}{5}\]

By following these steps, you can solve for the desired variable through systematic algebraic manipulation.

Understanding linear relationships is crucial when solving linear equations. A linear equation represents a straight-line relationship between two variables. In this exercise, the given equation \[5x + 2y = 9x - 3y\] showcases a linear relationship between \(x\) and \(y\).

Once we solve the equation for \(y\), the result, \[y = \frac{4x}{5}\], indicates a direct relationship between \(y\) and \(x\).

Key points about linear relationships:

• The relationship is direct; changes in \(x\) lead to proportional changes in \(y\).
• The equation can be represented graphically by a straight line on a coordinate plane.
• The slope is determined by the coefficient of \(x\), which in this case is \(\frac{4}{5}\).

Understanding these relationships helps in visualizing and solving linear equations more effectively.