Solving the Equation ab-ab-1: A Step-by-Step Guide

Understanding how to solve algebraic equations is a fundamental skill in mathematics. In this article, we will explore a detailed solution to the polynomial equation ab-ab-1. We will delve into the process of factoring, grouping, and substitution, making it easier for you to grasp the underlying concepts.

Factoring and Grouping Method

Let's begin by rewriting the given equation: ab - ab - 1. The first step is to factor the greatest common factor (GCF) from the first two terms:

ab - ab - 1  a1 - b1 - 1

Next, let's rewrite b - 1 as -1 - b to facilitate further grouping:

ab - ab - 1  a1 - b - 1 - b

Now, let's factor the common factor 1 - b from both terms:

ab - ab - 1  (1 - b) (a - 1)

The equation can now be written as:

ab - ab - 1  a - 1 - b

Hence, the correct choice is Option B: 1 - b a - 1.

Alternative Method Using Substitution

Another effective approach to solving the equation is by using substitution. Let's test the given choices with specific values.

Substitution Verification

Choose the values a 1 and b 0. Evaluating the expression ab - ab - 1 with these values:

ab - ab - 1  1*0 - 1*0 - 1  -1

The given choices are:

A: a1 b - 1 0 * 0 - 1 -1 B: 1 - b a - 1 1 - 0 * 1 - 1 0 C: a - 1 b1 1 - 1 * 0 * 1 0 D: a1 - b b - 1 1 - 0 * 1 - 1 -1

From the above, we can exclude choice A and D, as they do not satisfy the equation. Next, choose the values a 0 and b 1. Evaluating the expression ab - ab - 1 with these values:

ab - ab - 1  0*1 - 0*1 - 1  -1

The given choices are:

B: 1 - b a - 1 1 - 1 * 0 - 1 0 C: a - 1 b1 0 - 1 * 1 * 1 -2

From the above, we can exclude choice C, as it does not satisfy the equation. The only option remaining is B

Final Verification

We have verified that:

1 - b a - 1  1 - 0 * 1 - 1  0

And:

a - 11 - b  0 - 1 * 1 - 1  0

This confirms that the correct answer is indeed Option B: 1 - b a - 1.

In conclusion, by using both factoring and substitution methods, we have shown that the equation ab - ab - 1 simplifies to a - 1 - b1, making the correct choice B.

References and Further Reading

To deepen your understanding of algebraic equations and factoring techniques, refer to:

Algebra I For Dummies by Mary Jane Sterling Solving Algebraic Equations (Khan Academy)

These resources will provide you with a comprehensive understanding of the concepts involved.