Proving and Understanding the Algebraic Equation ( ab c ab - c cdot c ) for Real Numbers

When working with algebraic equations involving real numbers, it is crucial to understand the conditions under which specific equations hold true. Let's explore the equation ab c ab - c cdot c to determine under what conditions this equation is true.

Step-by-Step Proof

To begin with, let's start from the right-hand side (RHS) of the equation and manipulate it step by step:

RHS: (ab - c cdot c)

First, distribute (a):

RHS: (ab - a cdot c cdot c)

Next, we can rewrite it as:

RHS: (ab - ac cdot c)

Rearrange the terms to match the left-hand side (LHS):

RHS: (ab - ac cdot c c)

Now, let's compare this with the left-hand side of the equation:

LHS: (ab c)

To make the RHS equal to the LHS, we need:

RHS: (ab - ac cdot c c ab c)

For this to be true, the term (-ac cdot c) must also equal (c). This can be simplified as:

(-ac cdot c c)

This equation holds true if either:

a 0 c 0

Thus, the equation (ab c ab - c cdot c) is not universally true for all real numbers (a), (b), and (c). It holds true specifically when either (a 0) or (c 0).

Explanation and Implications

Understanding that the equation is valid only under specific conditions is crucial. Let's break down the implications:

When (a 0): The equation simplifies to (0 cdot b c 0 cdot b - 0 cdot b cdot c), which is true for any value of (b) and (c). When (c 0): The equation simplifies to (ab 0 ab - 0 cdot 0), which is also true for any values of (a) and (b).

However, if neither (a 0) nor (c 0), the equation does not hold true. This highlights the importance of checking the conditions under which algebraic manipulations are valid.

Conclusion

Therefore, we can conclude that the equation (ab c ab - c cdot c) is only valid for real numbers (a), (b), and (c) when either (a 0) or (c 0). It is crucial to verify such algebraic equations under different conditions to ensure their correctness.

To summarize:

The equation (ab c ab - c cdot c) is not universally true for all real numbers (a), (b), and (c). It is true specifically when either (a 0) or (c 0).

Understanding these conditions is fundamental in algebra and can greatly enhance one's problem-solving skills.