91Ó°ÊÓ

When it comes to math, linear equations are one of the most fundamental concepts. A linear equation is an equation where the highest power of the variable is one. These equations form a straight line when graphed, which is why they are called 'linear.' For instance, the equation provided in our exercise, \(5x - 4y = 2\), is a classic example. Here, both variables, \(x\) and \(y\), are raised to the power of one. This makes it a linear equation. In general, linear equations can be written in the form \(Ax + By = C\), where A, B, and C are constants.

Rearranging equations is a powerful technique in algebra that helps us isolate a specific variable. This skill is fundamental for solving equations. In our exercise, we need to solve for the variable \(y\). To do this, we rearrange the given equation. Starting with \(5x - 4y = 2\), the goal is to get \(y\) by itself. We can do this by performing inverse operations to move terms involving other variables or constants to the other side of the equation. Here are the steps we followed:
- First, subtract \(5x\) from both sides to isolate the term with \(y\): \[-4y = 2 - 5x\]
- Next, divide both sides by \(-4\) to solve for \(y\): \[y = \frac{2 - 5x}{-4} \]

Algebraic manipulation involves using mathematical operations to simplify or solve equations. It is a critical skill for working with all types of equations. In the given exercise, after isolating \(y\), we used algebraic manipulation to simplify the expression even further. Here's how we did it:
- We began with \(y = \frac{2 - 5x}{-4}\).
- Next, we separated the terms in the numerator to: \[y = \frac{2}{-4} - \frac{5x}{-4} \].
- Each fraction was then simplified: \[y = -\frac{1}{2} + \frac{5x}{4}\]
- Finally, we rearranged the terms to have \( x \) first: \[y = \frac{5x}{4} - \frac{1}{2}\].
Through these steps, we used both fundamental arithmetic and understanding of negative operations to simplify the equation successfully.