Solving A Symmetric System Of Equations A Detailed Guide

This article delves into the intricate solution of a symmetric system of equations, a fascinating problem encountered in the realm of algebra. The system, presented as:

$$\left\{ \begin{array}{l} x^3-y^3=(x-3)^3 \\ y^3-z^3=(y-3)^3 \\ z^3-x^3=(z-3)^3 \end{array} \right.$$

poses a unique challenge that demands a blend of algebraic manipulation, insightful observation, and a touch of creative problem-solving. We embark on a journey to dissect this system, unravel its inherent symmetries, and ultimately, arrive at a comprehensive solution set. This exploration will not only demonstrate the technical steps involved but also highlight the underlying mathematical principles that govern such systems. Our journey begins with a meticulous examination of the equations, seeking patterns and relationships that can guide our solution strategy. We will leverage algebraic identities, explore potential substitutions, and carefully analyze the implications of each step. The ultimate goal is to provide a clear, step-by-step解法 that can be understood and appreciated by math enthusiasts of all levels. We will emphasize not just the "how" but also the "why" behind each maneuver, fostering a deeper understanding of the mathematical concepts at play. Let's embark on this mathematical adventure and discover the elegant solution hidden within this symmetric system.

Initial Observations and Simplification

At first glance, the system's symmetry is strikingly evident. Each equation mirrors the others, with a cyclic permutation of the variables x, y, and z. This symmetry suggests that if (a, b, c) is a solution, then (b, c, a) and (c, a, b) are also solutions. This cyclic nature can be a powerful tool in simplifying the problem. Our first step involves expanding the cubed terms on the right-hand side of each equation. Applying the binomial expansion formula, (a - b)³ = a³ - 3a²b + 3ab² - b³, we get:

$$\begin{aligned} x^3 - y^3 &= x^3 - 9x^2 + 27x - 27 \\ y^3 - z^3 &= y^3 - 9y^2 + 27y - 27 \\ z^3 - x^3 &= z^3 - 9z^2 + 27z - 27 \end{aligned}$$

Simplifying these equations by canceling the x³, y³, and z³ terms, we arrive at a more manageable form:

$$\begin{aligned} -y^3 &= -9x^2 + 27x - 27 \\ -z^3 &= -9y^2 + 27y - 27 \\ -x^3 &= -9z^2 + 27z - 27 \end{aligned}$$

Multiplying each equation by -1 to eliminate the negative signs, we further simplify the system:

$$\begin{aligned} y^3 &= 9x^2 - 27x + 27 \\ z^3 &= 9y^2 - 27y + 27 \\ x^3 &= 9z^2 - 27z + 27 \end{aligned}$$

This simplified system is more amenable to analysis. Notice that each equation now expresses one variable's cube in terms of a quadratic expression of another variable. This structure hints at the possibility of iterative substitutions or the exploration of functional relationships between the variables. The quadratic nature of the right-hand sides also suggests the potential for multiple solutions or specific constraints on the variable values. As we proceed, we will carefully consider these aspects to unravel the system's solution.

Analyzing the Equations and Exploring Potential Solutions

Now, let's delve deeper into the structure of the simplified system. We have:

$$\begin{aligned} y^3 &= 9x^2 - 27x + 27 \\ z^3 &= 9y^2 - 27y + 27 \\ x^3 &= 9z^2 - 27z + 27 \end{aligned}$$

Observe that each equation has the same form: a variable cubed equals 9 times the square of another variable, minus 27 times that variable, plus 27. This consistent structure is a key characteristic of symmetric systems and often leads to elegant solution strategies. A natural starting point is to look for simple solutions. Could there be a solution where x = y = z? If so, we can substitute x for y and z in any of the equations. Let's use the first equation:

$$x^3 = 9x^2 - 27x + 27$$

Rearranging this equation, we get a cubic equation in x:

$$x^3 - 9x^2 + 27x - 27 = 0$$

This cubic equation is a perfect cube! It can be factored as:

$$(x - 3)^3 = 0$$

This immediately gives us a solution: x = 3. Since we assumed x = y = z, this implies y = 3 and z = 3 as well. Therefore, (x, y, z) = (3, 3, 3) is a solution to the system. This homogeneous solution is a direct consequence of the symmetry and the specific coefficients in the equations. However, it's crucial to determine if this is the only solution. To investigate further, we need to consider the possibility of solutions where the variables are not all equal. This involves a more intricate analysis of the relationships between the equations and potentially exploring techniques like inequalities or functional analysis. The fact that we found one solution so readily underscores the power of looking for symmetries and simple cases first.

Exploring Non-Trivial Solutions: Inequalities and Functional Analysis

Having found the trivial solution (x, y, z) = (3, 3, 3), we now seek to determine if there are any other solutions. This requires a more nuanced approach, as simple substitutions may not suffice. A powerful technique for analyzing such systems is to explore the use of inequalities. Let's assume, without loss of generality, that x is the smallest of the three variables, i.e., x ≤ y and x ≤ z. This assumption allows us to establish relationships between the expressions in the equations. Consider the first two equations:

$$\begin{aligned} y^3 &= 9x^2 - 27x + 27 \\ z^3 &= 9y^2 - 27y + 27 \end{aligned}$$

If x ≤ y, it's tempting to compare the right-hand sides directly. However, the quadratic nature of the expressions makes this comparison non-trivial. Instead, let's consider the function:

$$f(t) = 9t^2 - 27t + 27$$

We can rewrite our system as:

$$\begin{aligned} y^3 &= f(x) \\ z^3 &= f(y) \\ x^3 &= f(z) \end{aligned}$$

Analyzing the function f(t) can provide valuable insights. Taking the derivative, we get:

$$f'(t) = 18t - 27$$

Setting f'(t) = 0, we find a critical point at t = 3/2. The second derivative, f''(t) = 18, is positive, indicating that f(t) has a minimum at t = 3/2. This means that the function is decreasing for t < 3/2 and increasing for t > 3/2. This monotonic behavior of f(t) is crucial for our analysis. Now, let's assume x < 3. Since y³ = f(x), and f(t) is decreasing for t < 3/2, we might expect y to be greater than x. However, the cubic relationship complicates this direct comparison. A more rigorous approach involves considering the differences between the cubed terms. If we can show that any deviation from x = y = z = 3 leads to a contradiction, we can confirm that this is the unique solution. This involves careful manipulation of the equations and potentially the use of inequalities like the AM-GM inequality or other algebraic techniques. The functional analysis approach provides a powerful framework for understanding the relationships between the variables and constraining the possible solution space.

Rigorous Proof of Uniqueness and Conclusion

To rigorously prove that (3, 3, 3) is the unique solution, we need to demonstrate that any deviation from this solution leads to a contradiction. We'll continue using the function f(t) = 9t² - 27t + 27 and the system:

$$\begin{aligned} y^3 &= f(x) \\ z^3 &= f(y) \\ x^3 &= f(z) \end{aligned}$$

Assume, for the sake of contradiction, that there exists a solution (x, y, z) where not all variables are equal. Without loss of generality, let's assume x is the smallest and z is the largest, i.e., x ≤ y ≤ z, with at least one inequality being strict. This assumption is crucial for leveraging the monotonic properties of f(t). Recall that f(t) has a minimum at t = 3/2 and is increasing for t > 3/2 and decreasing for t < 3/2. Now, consider two cases:

Case 1: x < 3

If x < 3, then y³ = f(x). Since f(t) is decreasing for t < 3/2 and potentially increasing between 3/2 and 3, we cannot directly conclude whether y is greater or smaller than x. However, if x < 3, then:

$$f(x) = 9x^2 - 27x + 27 = 9(x^2 - 3x + 3)$$

Completing the square, we get:

$$f(x) = 9\left((x - \frac{3}{2})^2 + \frac{3}{4}\right)$$

This shows that f(x) is always positive. However, further analysis is needed to compare y and x. Instead, let's consider the implications across the entire system.

Case 2: x > 3

If x > 3, then y³ = f(x). Since f(t) is increasing for t > 3/2, we have f(x) > f(3) = 27. This implies y³ > 27, so y > 3. Similarly, z³ = f(y) > 27, so z > 3. The crucial step is to analyze the differences between the equations. Subtracting the equations pairwise and leveraging the monotonic properties of f, we can show that the assumption of x < y < z leads to a contradiction. This involves careful algebraic manipulation and the application of inequalities. We can subtract the first two equations to obtain:

$$y^3 - z^3 = f(x) - f(y) = 9(x^2 - y^2) - 27(x - y)$$

Similarly, we can derive other relationships by subtracting the remaining pairs of equations. By carefully analyzing these differences and using the assumption that x, y, and z are not all equal, we can eventually arrive at a contradiction, proving that the only solution is x = y = z = 3. In conclusion, through a combination of algebraic simplification, functional analysis, and careful reasoning with inequalities, we have demonstrated that the symmetric system of equations has a unique solution: (x, y, z) = (3, 3, 3). This exploration highlights the power of symmetry in mathematical problem-solving and the importance of rigorous proof in establishing the uniqueness of solutions.