Understanding and Identifying Counterexamples in Algebraic Expressions: (frac{aba-b}{a-b} ≠ ab)

When dealing with algebraic expressions, it is crucial to understand when certain equations hold true and when they do not. In this article, we will explore the algebraic expression (frac{aba-b}{a-b} ab) and identify a counterexample to demonstrate when this equation does not hold true.

Introduction to the Equation

The given equation is (frac{aba-b}{a-b} ab). To find a counterexample, we need to identify conditions under which this equation might not hold true. Let's first simplify the left-hand side of the expression.

Simplifying the Expression

The left-hand side of the equation simplifies to (ab) when (a - b eq 0). However, if (a - b 0), the expression becomes undefined because division by zero is not allowed in mathematics. Therefore, we need to consider the case where (a b).

Constructing a Counterexample

A simple counterexample can be constructed by choosing values for (a) and (b) such that (a b). Let's choose:

(a 2) (b 2)

Substituting these values into the expression, we get:

[frac{2 cdot 2 - 2}{2 - 2} frac{4 - 2}{0} frac{2}{0}]

This expression is undefined because division by zero is not permissible. Thus, the equation (frac{aba-b}{a-b} ab) does not hold when (a b).

Conclusion

The counterexample (a 2) and (b 2) demonstrates that the equation (frac{2 cdot 2 - 2}{2 - 2} eq 2 cdot 2) since it results in undefined division. This example highlights the importance of considering the conditions under which an expression is valid.

Further Examples and Discussion

Let's consider a more specific example to further illustrate this concept. If you let (a b 5), then:

[frac{5 cdot 5 - 5}{5 - 5} frac{25 - 5}{0} frac{20}{0}]

This again results in an undefined value, confirming that the equation does not hold in this case.

Therefore, we can conclude that the equation (frac{aba-b}{a-b} ab) is not valid when (a b) since division by zero is not possible.