Solving Systems of Equations for Real Numbers

When dealing with systems of equations, one common approach is to find the values of the variables that satisfy all the given equations simultaneously. In this article, we tackle a specific problem involving a system of equations concerning real numbers a, b, and c. The goal is to find all real numbers that satisfy the following system:

[begin{cases}2a - b a^2b 2b - c b^2c 2c - a c^2aend{cases}]

Step-by-Step Solution

First Equation:

[2a - b a^2b]

Rearranging the equation:

[b frac{2a}{1 - a^2}]

Second Equation: Substituting [b frac{2a}{1 - a^2}] into the second equation:

[2b - c b^2c]

Substitution:

[c frac{2b}{1 - b^2}]

Now, expressing [c] in terms of [a]:

[c frac{2 cdot frac{2a}{1 - a^2}}{1 - left(frac{2a}{1 - a^2}right)^2}]

Third Equation: Substituting the expression for [c] into the third equation:

[2c - a c^2a]

Substitution:

[a frac{2c}{1 - c^2}]

Now, expressing [a] in terms of itself:

[a frac{2 cdot frac{2 cdot frac{2a}{1 - a^2}}{1 - left(frac{2a}{1 - a^2}right)^2}}{1 - left(frac{2 cdot frac{2a}{1 - a^2}}{1 - left(frac{2a}{1 - a^2}right)^2}right)^2}]

Special Cases

To find all real solutions, we should explore special cases where certain variables are zero:

Case a 0:

First equation: [2(0) - b (0)^2b Rightarrow b 0] Second equation: [2(0) - c (0)^2c Rightarrow c 0] Third equation: [2(0) - 0 (0)^2(0) Rightarrow 0 0]

Solution: 0 0 0

Case b 0:

First equation: [2a - 0 a^2(0) Rightarrow 2a 0 Rightarrow a 0] Second equation: [2(0) - c (0)^2c Rightarrow c 0]

Solution: 0 0 0

Case c 0:

Third equation: [2c - a c^2a Rightarrow 2(0) - a (0)^2a Rightarrow -a 0 Rightarrow a 0] Second equation: [2a - 0 a^2(0) Rightarrow 2a 0 Rightarrow a 0]

Solution: 0 0 0

Exploring non-trivial solutions, where [a, b, c eq 0], we can substitute [b frac{2a}{1 - a^2}] into the second equation:

[2 cdot frac{2a}{1 - a^2} - c left(frac{2a}{1 - a^2}right)^2 c]

Simplifying the equation, we can solve for [c].

Observing patterns, if [a b c], then substituting into the first equation:

[2a - a a^3 Rightarrow a 0 text{ or } a 1]

Verifying the other equations with the same solution: [a 0 text{ or } a 1].

Thus, the solutions to the system are:

[a b c 0 text{ and } a b c 1]