Solving a Complex Equation through Algebraic and Computational Methods

Equations involving multiple variables and high-degree terms can be challenging to solve. In this article, we will walk through the process of solving the equation:

Equation Transformation

The given equation is:

y - x 3 - 2 x 6 80 [/math]

We can start by manipulating this equation to make it more manageable. One approach is to isolate the square term:

y - x 3 - 2 x 6 80 y - x 3 - 2 x 6

Step-by-Step Solution

Let's consider the equation:

y - x 3 - 2 x 6 80

From this, we can express the quadratic term as:

y - x 3 - 2 x 6 80 y - 2 x 6 - x 3

By substituting ( y x^3 k ), where ( k ) is a constant, we can solve for ( x ) and ( y ). We need ( x ) to be a real number such that the entire expression equals 80. Let's test the possible values:

x 6 80 x plusmn; 1 , 2

Since ( x^6 leq 80 ), only ( x pm 1 ) and ( x pm 2 ) are relevant. Now, let's substitute these values to find ( y ).

Cases for ( x )

Case 1: ( x 1 )

y - x 3 - 2 x 6 y - x - 2 x 6 80 y - - x - 2 80 y - - x - 2 80 y x - 2 - 4 y x - 2 12

Case 2: ( x 2 )

y - x 3 - 2 x 6 y - 8 - 32 80 y - 40 80 y 4

Similarly, we can solve for ( x -2 ) and ( x -1 ).

Conclusion

The solution set for the equation ( y - x^3 - 2x^6 80 ) is:

For ( x 1 ): ( y -4 ) and ( y 12 ) For ( x 2 ): ( y 4 ) For ( x -2 ): ( y -12 ) and ( y -4 ) For ( x -1 ): (Not applicable as it does not satisfy the equation)

Algebraic and Computational Methods

Using algebraic methods, we have solved the equation. However, for a more general and comprehensive solution, computational methods, particularly using a Computer Algebra System (CAS) like Mathematica, can be employed. Here’s how:

Using Mathematica, we can input the following code to solve the equation:

Reduce[2 x^6 y^2 - 2 x^3 y 80 , {x, y}]

This will yield the solution set as: {'x -2 land y -12 lor x -2 land y -4 lor x 2 land y 4 lor x 2 land y 12'}

To find the solution for integers, we use:

Reduce[2 x^6 y^2 - 2 x^3 y 80 , {x, y}, Integers]

The result is: {%2 -2 land y -12 lor x -2 land y -4 lor x 2 land y 4 lor x 2 land y 12%}

For a general equation (a x^6 b y^2 - c x^3 y d), the general solution can be found as:

y - c x 3 #xB1; - 4 a b x 6 - 4 b d c 2 x 6 2 b

with the condition (b eq 0).

By understanding both algebraic and computational methods, we can effectively solve complex equations and verify our solutions accurately.