Solving 8x over 2y when 3xy12

Investigating the algebraic relationship between the variables x and y, expressed as an equation 3xy 12, we aim to solve for the value of 8x / 2y. This article delves into the algebraic manipulation required to simplify and solve this equation, providing a clear, step-by-step explanation.

Keywords: exponential equations, algebraic manipulation, logarithms

Understanding the given equation

We are provided with the equation:

3xy 12 ………. 1

The task is to determine the value of 8x / 2y.

Step-by-step Solution

Step 1: Express 8x in terms of base 2

Firstly, we know that 8 23. Therefore, 8x can be rewritten as (23)x 23x.

Step 2: Rewrite the expression using the same base

Now, let's rewrite the expression using the same base:

8x / 2y 23x / 2y

Using the properties of exponents, we can subtract the exponents:

23x / 2y 23x - y

Step 3: Determine the value of 3x - y

From the given equation 3xy 12, we can solve for 3x - y:

3x - y 12 ……. 1

Substitute the value of 3x - y in the expression:

23x - y 212

Step 4: Calculate the final value

Since 212 4096, the value of 8x / 2y is 4096.

Conclusion

The solution to the problem is straightforward when we express 8x as 23x and then use the properties of exponents to simplify the expression. By using the given equation 3xy 12, we find that 3x - y 12, and hence 8x / 2y 212 4096.

Mathematical Insight

This problem embodies the principles of exponential equations and algebraic manipulation, providing valuable practice for students and professionals in mathematics. Understanding these concepts is essential for anyone dealing with complex algebraic expressions.

Mathematical Insight: Exponential equations and algebraic manipulation are crucial in a variety of fields, including computer science, physics, and engineering.