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Remember that you plan to respond to 12 questions and hope to score 600 points to win. It's possible to score even more points or many fewer (including negative scores) on the show.

Here are the equations you wrote to model your winning scenario:
[tex]\[
\begin{array}{l}
x + y = 12 \\
100x - 200y = 600
\end{array}
\][/tex]

Are the boundaries of the first equation viable in the second equation? What does this suggest about your plan to score 600 points?

Select the two correct statements.

A. The boundaries of the first equation result in scores of [tex]$-2,400$[/tex] and 1,200, which are both viable.
B. The lower boundary of the first equation is not viable because there can't be negative scores.
C. The upper boundary of the first equation results in a score less than 600 points, so this solution isn't viable and won't win the game.
D. Because 1,200 is greater than 600, there is probably a way to score 600 points and win even with a wrong answer or two.



Answer :

GinnyAnswer
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.

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