Let X X X , Y Y Y , And Z Z Z Be Positive Integers Such That ( X + Y + Z + 1 ) ( X Y + Y Z + Z X + X + Y + Z + 1 ) = X Y Z + 2023 (x + Y + Z + 1)(xy + Yz + Zx + X + Y + Z + 1) = Xyz + 2023 ( X + Y + Z + 1 ) ( X Y + Yz + Z X + X + Y + Z + 1 ) = X Yz + 2023 . Find X Y + Y Z + Z X . Xy + Yz + Zx. X Y + Yz + Z X .

$$$$$$$$$$

Introduction

In this problem, we are given three positive integers $x$, $y$, and $z$ that satisfy the equation $(x + y + z + 1)(xy + yz + zx + x + y + z + 1) = xyz + 2023$. Our goal is to find the value of $xy + yz + zx$.

The Given Equation

The given equation is $(x + y + z + 1)(xy + yz + zx + x + y + z + 1) = xyz + 2023$. This equation involves the sum of the three integers $x$, $y$, and $z$, as well as their pairwise products and the sum of their pairwise products.

Expanding the Left-Hand Side

To begin solving this problem, we can expand the left-hand side of the given equation. This will allow us to simplify the equation and potentially isolate the term $xy + yz + zx$.

(x + y + z + 1)(xy + yz + zx + x + y + z + 1) = x^2y + xy^2 + x^2z + xz^2 + y^2z + yz^2 + x^2 + xy + xz + y^2 + yz + z^2 + 2xy + 2xz + 2yz + x + y + z + 1

Simplifying the Equation

Now that we have expanded the left-hand side of the equation, we can simplify it by combining like terms.

x^2y + xy^2 + x^2z + xz^2 + y^2z + yz^2 + x^2 + xy + xz + y^2 + yz + z^2 + 2xy + 2xz + 2yz + x + y + z + 1 = xyz + 2023

Rearranging the Terms

To make it easier to isolate the term $xy + yz + zx$, we can rearrange the terms in the equation.

x^2y + xy^2 + x^2z + xz^2 + y^2z + yz^2 + x^2 + xy + xz + y^2 + yz + z^2 + 2xy + 2xz + 2yz + x + y + z + 1 - xyz - 2023 = 0

Factoring the Equation

Now that we have rearranged the terms, we can factor the equation to make it easier to solve.

(x + y + z + 1)(xy + yz + zx + x + y + z + 1) - xyz - 2023 = 0

Isolating the Term $xy + yz + zx$

To find the value of $xy + yz + zx$, we can isolate this term in the equation.

(xy + yz + zx + x + y + z + 1) = \frac{xyz + 2023}{x + y + z + 1}

Simplifying the Equation

Now that we have isolated the term $xy + yz + zx$, we simplify the equation by combining like terms.

xy + yz + zx = \frac{xyz + 2023}{x + y + z + 1} - (x + y + z + 1)

Using the Given Equation

We can use the given equation to substitute for $xyz + 2023$ in the equation above.

xy + yz + zx = \frac{(x + y + z + 1)(xy + yz + zx + x + y + z + 1) - (x + y + z + 1)}{x + y + z + 1}

Simplifying the Equation

Now that we have substituted for $xyz + 2023$, we can simplify the equation by combining like terms.

xy + yz + zx = \frac{(x + y + z + 1)(xy + yz + zx + x + y + z + 1) - (x + y + z + 1)}{x + y + z + 1}

Canceling the Common Factor

We can cancel the common factor of $x + y + z + 1$ in the numerator and denominator.

xy + yz + zx = xy + yz + zx + x + y + z + 1 - 1

Simplifying the Equation

Now that we have canceled the common factor, we can simplify the equation by combining like terms.

xy + yz + zx = x + y + z + 1

Using the Given Equation

We can use the given equation to substitute for $x + y + z + 1$ in the equation above.

xy + yz + zx = \sqrt{xyz + 2023}

Simplifying the Equation

Now that we have substituted for $x + y + z + 1$, we can simplify the equation by combining like terms.

xy + yz + zx = \sqrt{xyz + 2023}

Finding the Value of $xy + yz + zx$

To find the value of $xy + yz + zx$, we can substitute the given value of $xyz + 2023$ into the equation above.

xy + yz + zx = \sqrt{2023}

Simplifying the Equation

Now that we have substituted for $xyz + 2023$, we can simplify the equation by combining like terms.

xy + yz + zx = \sqrt{2023}

Evaluating the Square Root

To find the value of $xy + yz + zx$, we can evaluate the square root of $2023$.

xy + yz + zx = 44.8

Rounding the Answer

Since $xy + yz + zx$ must be an integer, we can round the answer to the nearest integer.

xy + yz + zx = 45

Conclusion

In this problem, we were given three positive integers $x$, $y$, and $z$ that satisfy the equation $(x + y + z + 1)(xy + yz + zx + x + y + z + 1) = xyz + 2023$. Our goal was to find the value of $xy + yz + zx$. We were able to isolate this term in the equation and simplify it to find the value of $ + yz + zx$. The final answer is $\boxed{45}$.

$$$$$$$$$$

Q: What is the given equation in the problem?

A: The given equation is $(x + y + z + 1)(xy + yz + zx + x + y + z + 1) = xyz + 2023$.

Q: What is the goal of the problem?

A: The goal of the problem is to find the value of $xy + yz + zx$.

Q: How can we start solving the problem?

A: We can start by expanding the left-hand side of the given equation.

Q: What is the expanded form of the left-hand side of the equation?

A: The expanded form of the left-hand side of the equation is $x^2y + xy^2 + x^2z + xz^2 + y^2z + yz^2 + x^2 + xy + xz + y^2 + yz + z^2 + 2xy + 2xz + 2yz + x + y + z + 1$.

Q: How can we simplify the equation?

A: We can simplify the equation by combining like terms.

Q: What is the simplified form of the equation?

A: The simplified form of the equation is $x^2y + xy^2 + x^2z + xz^2 + y^2z + yz^2 + x^2 + xy + xz + y^2 + yz + z^2 + 2xy + 2xz + 2yz + x + y + z + 1 - xyz - 2023 = 0$.

Q: How can we isolate the term $xy + yz + zx$?

A: We can isolate the term $xy + yz + zx$ by rearranging the terms in the equation.

Q: What is the isolated form of the term $xy + yz + zx$?

A: The isolated form of the term $xy + yz + zx$ is $xy + yz + zx = \frac{xyz + 2023}{x + y + z + 1} - (x + y + z + 1)$.

Q: How can we simplify the equation further?

A: We can simplify the equation further by using the given equation to substitute for $xyz + 2023$.

Q: What is the simplified form of the equation?

A: The simplified form of the equation is $xy + yz + zx = \frac{(x + y + z + 1)(xy + yz + zx + x + y + z + 1) - (x + y + z + 1)}{x + y + z + 1}$.

Q: How can we cancel the common factor in the numerator and denominator?

A: We can cancel the common factor of $x + y + z + 1$ in the numerator and denominator.

Q: What is the simplified form of the equation after canceling the common factor?

A: The simplified form of the equation after canceling the common factor is $xy + yz + zx = x + y + z + 1$.

Q: How can we find the value of $xy + yz + zx$?

A: We find the value of $xy + yz + zx$ by using the given equation to substitute for $x + y + z + 1$.

Q: What is the value of $xy + yz + zx$?

A: The value of $xy + yz + zx$ is $\sqrt{2023}$.

Q: How can we evaluate the square root of $2023$?

A: We can evaluate the square root of $2023$ to find the value of $xy + yz + zx$.

Q: What is the value of $xy + yz + zx$ after evaluating the square root?

A: The value of $xy + yz + zx$ after evaluating the square root is $44.8$.

Q: How can we round the answer to the nearest integer?

A: We can round the answer to the nearest integer to find the final value of $xy + yz + zx$.

Q: What is the final value of $xy + yz + zx$?

A: The final value of $xy + yz + zx$ is $45$.

Q: What is the conclusion of the problem?

A: The conclusion of the problem is that the value of $xy + yz + zx$ is $45$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{45}$.