Solving the Equation: (ab 2a-3b) and Its Applications

Let's delve into the intriguing and fascinating world of algebra, focusing on the equation ab 2a - 3b. This equation holds a special place in the realm of algebraic expressions, offering a unique insight into the relationship between variables a and b. In this article, we will explore the solution steps, applications, and related concepts.

Solution of the Equation

Consider the given equation: ab 2a - 3b. We need to find the values of a and b that satisfy this equation. Let's break down the process into clear steps:

Step 1: Isolate a and b terms Given: ab 2a - 3b Rearrange to: ab 3b 2a Step 2: Factor out b on the left side Factor: b(a 3) 2a Step 3: Solve for b Isolate b: b 2a / (a 3)

This means that for any given value of a, we can find the corresponding value of b using the formula b 2a / (a 3).

Substitution and Calculation

Let's apply the derived formula to the specific values given: 35 23 - 35 and 53 25 - 33.

For 35:

Given: 35 23 - 35 Calculate: 35 6 - 15 * 15 Simplify: 35 6 - 15 Result: 35 6

For 53:

Given: 53 25 - 33 Calculate: 53 10 - 9 * 15 Simplify: 53 10 - 9 Result: 53 16

Combining the results:

Therefore, 35 53 616 22

Applications of the Equation

The equation ab 2a - 3b, while seemingly abstract, has practical applications in various fields. These applications include:

Mathematics: This equation demonstrates the principles of algebraic manipulation and solving simultaneous equations. It helps in understanding the interplay between different algebraic terms. Science: In physics, such equations can be used to model relationships between quantities in scenarios like dynamics or thermodynamics. Engineering: In electrical engineering, this equation can be used to solve circuit analysis problems where voltages and currents need to be balanced. Economics: Understanding equations like this can help in modeling economic systems where supply and demand need to be balanced.

Conclusion

Through solving and understanding the equation ab 2a - 3b, we gain valuable insights into the algebraic manipulations and problem-solving techniques that are essential in many fields. The process of solving this equation, and the steps involved, provide a solid foundation for tackling more complex equations and problems in the future.

By applying this knowledge, one can improve their problem-solving skills, which are highly beneficial in academic and professional settings. Remember, the key to mastering algebra is practice and comprehension of the underlying principles.