Understanding the Nature of Linear Equations: A Case Study of 2x/3 - x - 3 4x

Solving and understanding linear equations is a fundamental skill in algebra, providing a basis for more advanced mathematical concepts. This article will delve into the specifics of the equation 2x/3 - x - 3 4x, proving that it is indeed a linear equation. We will explore the steps required to simplify the equation, its standard form, and the significance of its graphical representation.

The Equation: 2x/3 - x - 3 4x

Let's start with the equation:

2x 3 - x - 3 4x

At first glance, this equation might seem complex due to the presence of fractions and variables on both sides. However, we can simplify it to reveal its underlying structure.

Simplification of the Equation

To simplify the equation, we will follow these steps:

Multiply the entire equation by 3 to eliminate the fraction. Simplify and combine like terms. Rearrange the equation to isolate the variable on one side. Express the equation in its standard form.

Step 1: Multiplying the Equation by 3

First, we multiply the entire equation by 3:

3 ? 2x 3 - 3 ? x - 3 3 ? 4x

This simplifies to:

2x - 3x - 9 12x

Step 2: Combining Like Terms

Next, we combine the like terms:

2x - 3x 12x 9

This simplifies to:

- 1x 9 - 12x

Further simplifying, we get:

- 7x 9

Step 3: Isolating the Variable

To isolate the variable, we divide both sides by -7:

x - 9 7

This simplifies to:

x - 9 7

Step 4: Standard Form

The equation can now be expressed in its standard form:

- 7x - 9 0

This confirms that 2x/3 - x - 3 4x is indeed a linear equation.

Conclusion: Importance of Linear Equations

Understanding linear equations is crucial in various fields, including science, engineering, and economics. It is a simple yet powerful tool that allows us to model real-world phenomena. The graph of a linear equation is a straight line, and the solution to the equation can be used to determine key points on this line, such as where it intersects the axes.

In conclusion, the equation 2x/3 - x - 3 4x is a linear equation, and its solution x -9/7 is a clear example of the importance of algebraic simplification and the standard form in solving linear equations.