To solve this question, we need to consider the information provided in the equations and the resulting scores. Let's analyze step-by-step:
### Given Equations:
1. [tex]\( x + y = 12 \)[/tex]
2. [tex]\( 100x - 200y = 600 \)[/tex]
### Step-by-Step Solution:
1. Determine the boundaries of the first equation:
- When [tex]\( y = 0 \)[/tex]:
- [tex]\( x + 0 = 12 \)[/tex]
- Therefore, [tex]\( x = 12 \)[/tex]
- When [tex]\( x = 0 \)[/tex]:
- [tex]\( 0 + y = 12 \)[/tex]
- Therefore, [tex]\( y = 12 \)[/tex]
2. Evaluate the boundary conditions for these values in the second equation:
- When [tex]\( x = 0 \)[/tex] and [tex]\( y = 12 \)[/tex]:
- [tex]\( 100 \cdot 0 - 200 \cdot 12 = -2400 \)[/tex]
- When [tex]\( x = 12 \)[/tex] and [tex]\( y = 0 \)[/tex]:
- [tex]\( 100 \cdot 12 - 200 \cdot 0 = 1200 \)[/tex]
### Interpret the results:
1. The boundaries of the first equation result in scores of [tex]\(-2400\)[/tex] and [tex]\(1200\)[/tex]. This shows that:
- It is possible to achieve a score of [tex]\(1200\)[/tex], which is higher than the required [tex]\(600\)[/tex] points.
- A score of [tex]\(-2400\)[/tex] is possible, which is a negative score.
2. The negative score suggests that there is a scenario where a very low score is obtained, indicating the importance of the choices made.
3. Since a score of [tex]\(1200\)[/tex] is achievable, there is a way to score over [tex]\(600\)[/tex] points, giving some flexibility and indicating that even with some wrong answers, you might still score [tex]\(600\)[/tex] points.
### Correct Statements:
1. The boundaries of the first equation result in scores of [tex]\(-2,400\)[/tex] and [tex]\(1,200\)[/tex], which are both viable.
2. Because [tex]\(1,200\)[/tex] is greater than [tex]\(600\)[/tex], there is probably a way to score [tex]\(600\)[/tex] points and win even with a wrong answer or two.
These statements clarify that while high scores are possible, the strategy must be carefully planned to avoid negative scores.
