We are given two equations involving real numbers $x$ and $y$:

1. $xy = 9$
2. $(x^2)y + xyy + x + y = 100$

We are asked to find $x^2 + y^2$.

Step 1: Simplify the second equation

The second equation can be rewritten as: $(x^2)y + xyy + x + y = 100$ Notice that $x^2y + xyy = xy(x + y)$. So, the equation becomes: $xy(x + y) + x + y = 100$ Now, we substitute $xy = 9$ (from the first equation) into this expression: $9(x + y) + x + y = 100$ Factor out $(x + y)$: $(9 + 1)(x + y) = 100$ $10(x + y) = 100$ Solve for $x + y$: $x + y = \frac{100}{10} = 10$

Step 2: Solve for $x^2 + y^2$

We can use the identity: $x^2 + y^2 = (x + y)^2 - 2xy$ Substitute $x + y = 10$ and $xy = 9$ into this formula: $x^2 + y^2 = (10)^2 - 2(9)$ $x^2 + y^2 = 100 - 18$ $x^2 + y^2 = 82$

Final Answer:

$x^2 + y^2 = 82$

Would you like more details or have any questions about this process?

Related questions:

1. How can you derive the formula $x^2 + y^2 = (x + y)^2 - 2xy$?
2. What other strategies can be used to solve equations with two variables?
3. How can you solve quadratic equations that arise from the values of $x$ and $y$?
4. Can you solve the system if we were given different conditions, like $x + y = 5$?
5. How do you verify solutions for such systems of equations?

Tip: Always look for patterns or identities (like the difference of squares or the sum of squares) to simplify complicated expressions!