1. By: Nora R.

2. Problem Suppose x and y are real numbers such that xy = 9 and x²y + xy² + x + y = 100. What is the integer value of x² + y²?

3. First Step: The first thing I did was try to figure out what I had to multiply x and y by to get 9. I knew the product of x and y (xy) had to be 9 because it says in the problem that xy equals 9.

4. Second Step: I know that since xy equals 9 I have to multiply two numbers to get 9. So, I have two options, I can multiply 9×1 or 3×3. I guessed 9×1 first.

5. Third Step: So now what I did was solve the problem given (x²y + xy² + x + y) Since it doesn’t really matter what each variable is I just made x=9 and y=1 So, now the new problem is - 9²(1) + 9(1)² + 9 + 1 - and I want to see if this will equal 100.

6. Solving the Equation: To solve this problem I just figured out what each section equals and then added them up to see if it equaled 100. 9²(1) + 9(1)² + 9 + 1 = ? 9²= 81 81 × 1 = 81 In the order of operations exponents come before multiplication so I have to do the exponent before multiplying anything, which pretty much means I have to do 1² first, then multiply that by 9. (PEMDAS) 1²= 1 1×9= 9

7. Fourth Step: I added up all the numbers I got from the problem, to see if it equaled 100. When I added them up, it did equal 100, so I knew I had the right numbers for x and y. 81 9 1 + 100

8. Final Step: Now all I have to do to get my answer is solve the last problem giving with the variables I now have. (x=9, y=1) The problem asked for me to solve the problem – x² + y² So I replace the variables with 9 and 1 to find the answer. 9² + 1² 811 81 +1 = 82

9. Answer: Once I solved that problem of 9² + 1² I got and answer of 82.