Solving for (x^4 - y^4) Given (xy 13) and (x - y 3) Using Algebraic Techniques

In this article, we will explore how to solve a problem where we need to find the value of (x^4 - y^4) given the conditions (xy 13) and (x - y 3). This involves basic algebraic manipulation and the use of polynomial identities. Follow along to understand the step-by-step process.

Step-by-Step Solution:

Step 1: Solving for (x) and (y)

We start with the given equations:

[begin{cases}xy 13 x - y 3end{cases}]

To find the values of (x) and (y), we can add the two equations:

[xy (x - y) 13 3]

(2x 16)

Solving for (x):

[x frac{16}{2} 8]

Substituting (x 8) back into the equation (xy 13):

[8y 13 Rightarrow y frac{13}{8} 5]

Thus, the values of (x) and (y) are:

[x 8, quad y 5]

Step 2: Calculating (x^4 - y^4)

We use the identity for the difference of fourth powers:

[x^4 - y^4 (x^2 y^2)(x^2 - y^2)]

We first calculate (x^2) and (y^2):

[x^2 8^2 64, quad y^2 5^2 25]

Next, we calculate (x^2 - y^2):

[x^2 - y^2 64 - 25 39]

And (x^2 y^2):

[x^2 y^2 64 25 89]

Combining these results, we find:

[x^4 - y^4 (x^2 y^2)(x^2 - y^2) 89 times 39 3471]

Therefore, the value of (x^4 - y^4) is:

[boxed{3471}]

Generalized Solution Using Combined Equations:

Alternatively, we can directly use the combined equations to simplify the process:

[x^4 - y^4 (x^2 y^2)(x^2 - y^2)]

From the previous steps, we have:

[x^2 y^2 89 quad text{and} quad x^2 - y^2 39]

Substituting these values:

[x^4 - y^4 89 times 39 3471]

Conclusion:

This method can be applied to similar problems where you need to solve for polynomial expressions using given conditions. The key is to use algebraic identities and systematically solve for intermediate values.

Keywords: algebraic equations, simultaneous equations, polynomial expressions