Question

Solve Diophantus's Problem \(\mathrm{V}-10\) for the two given numbers 3,9

Short answer

Answer: The two numbers are x = 6 and y = 18.

Step-by-step solution

Rewrite the equations with variables

The problem statement has provided us with the two equations we need to work with. Let's rewrite them using the variables x and y: 1. x * y - 3 = (x + 3)^2 2. x * y - 9 = (y + 9)^2

Expand the equations

Now, we will expand the equations to make them easier to work with. The equations will then look like this: 1. x * y - 3 = x^2 + 6x + 9 2. x * y - 9 = y^2 + 18y + 81

Rearrange the equations

Rearrange the equations so that all the terms are on one side: 1. x * y - x^2 - 6x - 6 = 0 2. x * y - y^2 - 18y - 72 = 0

Solve the first equation for x

We'll now solve Equation 1 for x in terms of y. To do this, we can factor out x from the equation and then isolate x on one side: x * (y - 1) - 6 * (y - 1) = 0 (x - 6) * (y - 1) = 0 So, we have two possibilities: x = 6 or y = 1

Solve the second equation for y in terms of x

Now, we will solve Equation 2 for y in terms of x: x * y - y^2 - 18y - 72 = 0 Let's factor out y from the equation: y * (x - 1) - 18 * (y - 1) = 0 (y - 18) * (x - 1) = 0 So, we have two possibilities: y = 18 or x = 1

Find the values of x and y

For the final step, we'll use the possibilities derived in Steps 4 and 5 to find x and y: Possibility 1: x = 6 y = 18 Possibility 2: x = 1 y = 1 Since the problem asks to divide the given numbers into two other values, we will disregard the possibility of x = 1 and y = 1 because we're dividing the same numbers. Therefore, the solution is: x = 6, y = 18 Now we have found the two numbers that satisfy the given conditions of Diophantus's Problem V-10 for the numbers 3 and 9. The two numbers are 6 and 18.

Key concepts

Steps to Solve Equations

Diophantine equations present unique challenges, often requiring systematic approaches to find integer solutions. Follow these steps to effectively tackle such problems:

1. **Rewrite the Equations with Variables**: Substitute any given conditions or numbers in the problem into variable form to ease manipulation. For example, if given numbers 3 and 9, express them with variables like \( x \) and \( y \).
2. **Expand the Equations**: Transform the equations into polynomials if possible. This involves distributing any products and making sure all expressions are fully expanded.
3. **Rearrange the Equations**: Shift all terms to one side of the equation to prepare for further simplifications or factoring. This step is crucial for clarity and to identify potential solutions.

These steps lay the groundwork for finding solutions through systematic problem-solving strategies.

Algebraic Manipulation

Algebraic manipulation is a key tool in solving equations, especially when dealing with complex polynomials in Diophantine problems. The manipulation process involves several techniques:

• **Factoring**: Sometimes, factors are not immediately visible. Factoring expressions by grouping or using special identities can help simplify equations.
• **Simplification**: Leaving like terms uncombined can create unnecessary complication. Combine like terms to reduce clutter and solve equations smoothly.
• **Isolation of Variables**: This technique is essential for identifying values of unknowns. Rearrange terms to isolate one variable, which can simplify solving for the others.

By carefully manipulating algebraic expressions, we maintain equation balance and progress towards identifying the solution efficiently.

Problem-Solving in Mathematics

Mathematical problem-solving is both an art and a science, especially relevant for solving equations like Diophantine problems. It involves creativity and precision:

• **Identifying Patterns**: In problems involving integers, like Diophantine equations, look for repeating numerical or algebraic patterns to anticipate solutions.
• **Logical Deduction**: Use logic to infer solutions based on established possibilities, like considering the values found in prior steps.
• **Verification**: Always double-check the solutions within the context. Ensure they meet the problem's conditions and are logically consistent.

Through iterative and logical methods, we refine our approach to achieve clear solutions. This problem-solving approach ensures both accuracy and a deeper understanding of mathematical principles.