We are given the following system of equations for $a$ and $b$:

$a^2 + 121 = b^2 + 1369 = a^2 + b^2 - 2ab + 576$

We need to solve for $a$, $b$, and ultimately find the value of $ab$.

Step 1: Analyze the First Equality

From the first equation: $a^2 + 121 = b^2 + 1369$ Rearranging it: $a^2 - b^2 = 1369 - 121 = 1248$ This can be factored as: $(a - b)(a + b) = 1248$

Step 2: Analyze the Second Equality

From the second equation: $b^2 + 1369 = a^2 + b^2 - 2ab + 576$ Simplify it: $1369 = a^2 - 2ab + 576$ Rearrange it: $a^2 - 2ab = 1369 - 576 = 793$

Step 3: Solve the System

Now we have the system: $(a - b)(a + b) = 1248 \quad \text{(Equation 1)}$ $a^2 - 2ab = 793 \quad \text{(Equation 2)}$

We will solve this system step by step.

Want more details or clarification so far?

Here are 5 questions that expand the topic:

1. How can we solve for $a$ and $b$ using substitution?
2. What are the implications of factoring the difference of squares?
3. Could we find specific values of $a$ and $b$ using trial and error for integer solutions?
4. How does factoring simplify solving equations with higher powers?
5. What strategies can we use to simplify complex systems of equations?

Tip: When dealing with systems of equations, try to reduce the number of variables by substitution or elimination before solving.