Final answer:
The linear equation given can be used to find several combinations of multiple choice and free response questions. Three combinations are: (1) 100 multiple choice and 75 free response, (2) 200 multiple choice and 50 free response, and (3) 300 multiple choice and 25 free response, all yielding a total of 800 points.
Step-by-step explanation:
The equation given, 2x + 8y = 800, is a linear equation that models how many multiple choice questions (x) and how many free response questions (y) Amy needs to answer to earn 800 points. To find combinations that satisfy the equation, we can choose values for x and solve for y correspondingly, ensuring that both x and y are whole numbers, as you can't answer a fraction of a question.
- Combination 1: If Amy answers 100 multiple choice questions (x=100), then we have 2(100) + 8y = 800, which reduces to 200 + 8y = 800. Solving for y gives y = 75. So, one combination is 100 multiple choice and 75 free response questions.
- Combination 2: Choosing 200 multiple choice questions (x=200), the equation becomes 2(200) + 8y = 800, or 400 + 8y = 800. Solving for y yields y = 50. This combination is 200 multiple choice and 50 free response questions.
- Combination 3: With 300 multiple choice questions (x=300), the equation is 2(300) + 8y = 800, transforming into 600 + 8y = 800. The solution for y is 25. Therefore, another combination is 300 multiple choice and 25 free response questions.
Note that these combinations assume the correctness of scoring such that each multiple choice and each free response question contribute to the total score equally within their categories.