Solving the System of Equations: x3y224 and x2y368

In this article, we will explore how to solve the system of equations: and . This involves a series of algebraic manipulations to find the values of x and y that satisfy both equations.

Initial Equations

Let's start by writing down the given equations:

Assume Verification with Given Solutions

To verify, we substitute into the equations:

is true. is true.

We confirm that is a valid solution.

Manipulating the Equations Further

We can manipulate the equations to gain more insight:

Multiply the first equation by 6 and the second by 17 to get:

Subtract the first modified equation from the second:

This results in:

Since satisfy the original equations, we have:

Additionally, we find another solution:

However, this does not satisfy the original equations. Thus, the unique solution remains .

Verification of Uniqueness

To verify the uniqueness, we use the Arithmetic Mean-Geometric Mean (AM-GM) inequality:

Rearranging the above expression gives:

By AM-GM inequality:

This contradicts the earlier expression, proving the solution is unique.

In conclusion, the only solution to the system of equations is:

Validation: