Solving Simultaneous Equations Involving Integer Values

In this article, we delve into the process of finding positive integral values for variables (x) and (y) that satisfy the equations xy 9 and x^{1/3}y^{1/3} 3. This involves a systematic approach to solving these simultaneous equations, highlighting the importance of algebraic manipulation and the application of mathematical principles.

Problem Statement

We are given the following system of equations: xy 9 x^{1/3}y^{1/3} 3 Our goal is to determine if there exist positive integral values for (x) and (y) that satisfy these equations simultaneously.

Step-by-Step Solution

To solve the given system of equations, we start by expressing (y) in terms of (x) from the first equation:

y 9 - x

Substituting this expression into the second equation, we have:

x^{1/3}(9 - x^{1/3}) 3

Let's denote a x^{1/3} and b 9 - x^{1/3}. Therefore:

ab 3

a^3b^3 9

Using the identity a^3b^3 a(9 - a^{1/3})^{3/2} - a^{1/3}(9 - a^{1/3})^{1/2}, we can express (a^3b^3) in terms of (a) and (b).

Quadratic Equation Approach

Noting that (a^3b^3 9) and (ab 3), we can find (a^2 - ab b^2)

a^2b^2 9 - 2ab 9 - 6 3

From here, we can solve for (a^3b^3) as follows:

9 9 - 6ab 6ab 9 - 6(3) 6ab

ab 3

The resulting system of equations is:

ab 3

ab 3

These correspond to the roots of the quadratic equation:

t^2 - 3t 3 0

Using the quadratic formula:

t (3 ± sqrt{9 - 12}) / 2

These roots are complex:

t (3 ± isqrt{3})/2

This indicates that (a) and (b) cannot be real numbers, meaning there are no positive integral solutions (x, y) that satisfy both equations simultaneously.

Easier Method Using Substitution

We can make the process simpler by setting a x^{1/3} and b y^{1/3}. The given relations simplify to:

ab 3

a^3b^3 9

From 9 a^3b^3 ab(a^2 - ab b^2), we have:

9 3(9 - 6ab)

ab 2

Thus, solving ab 2 and ab 3 gives the solutions:

a 1, b 2

a 2, b 1

Converting back to (x) and (y), we get:

x 1, y 8

x 8, y 1

Therefore, the only positive integral values of (x) and (y) that satisfy the given equations are:

(1, 8) and (8, 1)

Thus, there are two pairs of positive integer solutions.

Conclusion

In conclusion, the process of solving simultaneous equations involves a combination of algebraic manipulation and careful substitution. The equations given in this article have positive integral solutions only when (x 1) and (y 8), or (x 8) and (y 1).