Solving the Equation 2x 5 3x - 4: Methods, Proofs, and Applications

The equation 2x 5 3x - 4 presents an interesting challenge in basic algebra. This article guides you through the process of solving the equation, providing a detailed step-by-step solution and a comprehensive proof of the solution's correctness.

Introduction to the Equation

Understanding and solving equations is fundamental in mathematics. The equation 2x 5 3x - 4 is a simple yet powerful example of a linear equation. Here, x is the variable we need to solve for.

Step-by-Step Solution

The first step is to simplify the equation and isolate the variable x on one side of the equation.

Method 1: Isolation Technique

1. Start with the Equation:

2x 5 3x - 4

2. Move the variable-terms to one side and numerical terms to the other:

Subtract 2x from both sides: 5 3x - 2x - 4

3. Simplify the equation:

5 x - 4 Add 4 to both sides: 5 4 x 9 x

Hence, x 9.

Method 2: Standard Algebraic Approach

1. Start with the Equation:

2x 5 3x - 4

2. Move the x-terms to one side and constant terms to the other:

Subtract 2x from both sides: 5 3x - 2x - 4

3. Simplify the equation:

5 x - 4 Add 4 to both sides: 5 4 x 9 x

This confirms that the variable x equals 9.

Proof of Solution

To ensure the accuracy of the solution, we substitute x 9 back into the original equation.

Original Equation:

2x 5 3x - 4

Substituting x 9:

2(9) 5 3(9) - 4 18 5 27 - 4 23 23

The equality 23 23 verifies that x 9 is the correct solution.

Further Exploration and Applications

The process of solving linear equations is not only theoretical but also has practical applications. Many real-world problems can be modeled using linear equations, such as economics, physics, and engineering.

Real-World Applications

For example, in economics, linear equations are used to model supply and demand curves. Understanding how to solve such equations is crucial for predicting market behaviors.

Conclusion

The equation 2x 5 3x - 4 is solved by isolating the variable x on one side of the equation. The solution x 9 has been verified through substitution, and the process can be applied to solve a wide range of similar equations in various fields of study.