91Ó°ÊÓ

Isolating variables is a core concept in algebra, allowing us to focus on one variable at a time. In the given problem, we need to solve for the variable \(y\). This means we want to express \(y\) in terms of \(x\).
First, we start by isolating the variable by rearranging the equation. For our example equation 2x - y = 1, we need to move all terms involving \(y\) to one side and other terms to the opposite side.
To achieve this, we subtract 2x from both sides of the equation:

2x - y - 2x = 1 - 2x

This yields:

-y = 1 - 2x

By doing so, we make our term \(-y\) the focal point of one side of the equation. This process is crucial because it helps simplify and prepare the equation for further manipulation.

Linear equations form the backbone of basic algebra. These equations have the general form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In our exercise, we are dealing with the linear equation 2x - y = 1.
Linear equations are characterized by their degree, which is 1, meaning that the variable's highest exponent is 1. They can be represented graphically by a straight line.
To solve this, after isolating the variable \(-y\), the next step involves algebraic manipulation to express \(y\). Multiply both sides of the equation by -1:-1(-y) = -1(1 - 2x)

This yields:

y = 2x - 1

Now, the equation is in its simplest linear form, showing a clear relationship between \(y\) and \(x\). Understanding linear equations is essential as they often serve as the basis for more complex algebraic problems.

Algebraic manipulation is the art of rearranging equations to solve for a desired variable. It involves various operations like addition, subtraction, multiplication, and division. Here’s how it works in our problem:
After we isolated \(-y\) on one side: \(-y = 1 - 2x\)
We needed to get \(y\) by itself. So, we multiplied both sides of the equation by -1 to eliminate the negative sign:

-1(-y) = -1(1 - 2x)

This simplifies to:
y = 2x - 1

Different algebraic manipulations like this are used to transform equations, making them simpler to solve. Key operations include:

• Adding or subtracting terms on both sides to isolate variables
• Multiplying or dividing both sides to cancel coefficients

By understanding these techniques, solving even the most challenging algebra problems can become straightforward.