We are given the system of equations:

1. $2x + 2y = 39$
2. $xy = 92$

Let's solve this system step by step.

Step 1: Simplify the First Equation

We can simplify the first equation by dividing both sides by 2: $x + y = \frac{39}{2} = 19.5$ So the simplified system of equations is:

1. $x + y = 19.5$
2. $xy = 92$

Step 2: Express $y$ in Terms of $x$

From the first equation $x + y = 19.5$, we can express $y$ as: $y = 19.5 - x$

Step 3: Substitute into the Second Equation

Now, substitute $y = 19.5 - x$ into the second equation $xy = 92$: $x(19.5 - x) = 92$

Expanding the equation: $19.5x - x^2 = 92$

Step 4: Form a Quadratic Equation

Rearrange the equation to standard quadratic form: $x^2 - 19.5x + 92 = 0$

Step 5: Solve the Quadratic Equation

Now, solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = -19.5$, and $c = 92$.

Let's calculate the roots of the equation.The solutions to the quadratic equation are:

$x = 11.5 \quad \text{or} \quad x = 8.0$

Step 6: Find the Corresponding Values of $y$

Using the equation $y = 19.5 - x$, we can find the corresponding values of $y$ for each solution of $x$:

1. If $x = 11.5$, then $y = 19.5 - 11.5 = 8.0$.
2. If $x = 8.0$, then $y = 19.5 - 8.0 = 11.5$.

Final Solution

The solutions to the system of equations are:

1. $(x, y) = (11.5, 8.0)$
2. $(x, y) = (8.0, 11.5)$

If you have any questions or need further details, feel free to ask!

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Here are some related questions:

1. How would you solve the system if the sum equation was different?
2. What happens if the product equation changes?
3. How does the quadratic formula apply to more complex cases?
4. What are other methods for solving systems of nonlinear equations?
5. Can you graphically represent the solutions?

Tip: Always check your solutions by substituting them back into the original equations to verify their correctness.