Solving Mathematical Expressions with Given Constraints

In this article, we will explore a detailed solution to an intriguing mathematical problem. We will break down the process step by step, explaining how to simplify a complex expression given certain constraints. This type of problem-solving technique is commonly used in various fields, from competitive mathematics to computer science.

Motivating Example: Expression Evaluation in Mathematical Context

Consider the expression:

2a-3 2b-3 2c-3 - 32a-2 2b-2 2c-2

Given the constraint that:

a ? b ? c 6

Step-by-Step Solution

The first step in solving such problems is to define new variables to simplify the given expression and the constraint. In this case, let's define the variables as follows:

Let

x 2 - a
y 2 - b
z 2 - c

Now, we can express a, b, and c in terms of x, y, and z:

a 2 - x
b 2 - y
c 2 - z

Substituting these into the given constraint:

(2 - x)(2 - y)(2 - z) 6

This simplifies to:

8 - 4x - 4y - 4z 2xy 2xz 2yz - xyz 6

Rearranging the terms, we get:

2 - x - y - z (xy xz yz - xyz)/2 3/4

However, for simplicity, we can directly substitute into the initial expression:

(2 - a)(2 - b)(2 - c)(2a-2 2b-2 2c-2 - 2 - a - 2 - b - 2 - c)

Given a * b * c 6, we know xyz 0 (since (2 - a)(2 - b)(2 - c) 6).

Algebraic Simplification

Now we need to simplify the expression:

x3 y3 z3 - 3xyz

Using the identity:

x3 y3 z3 - 3xyz (xyz)(x2 y2 z2 - x - y - z)

Since we know xyz 0, the entire expression simplifies to:

x3 y3 z3 - 3xyz 0 * (x2 y2 z2 - x - y - z) 0

Conclusion

Therefore, the value of the original expression is:

boxed{0}

Additional Practice

The given problem can be further generalized for practice. For example:

Let

2 - a x
2 - b y
2 - c z

Given that

abc 6

Then

xyz 0

Substituting these into the expression:

x3 y3 z3 - 3xyz, we get:

x3 y3 z3 - 3xyz 0 * x2 y2 z2 - xy - xz - yz 0

The final result remains:

boxed{0}

Summary

This article demonstrated how to evaluate a complex mathematical expression using algebraic identities and substitution methods. By transforming the given variables and applying known identities, we were able to simplify the expression to its final value.

The techniques outlined here are valuable for solving similar types of problems and are widely applicable in various mathematical and scientific contexts.