Which equation models the scoring in the final round and the outcome of the contest?

Steps to Solve

1. Identify the variables and constants in the question

In this problem, we will represent the number of correct answers given by one school as $x$, and note the total score for each school. According to the provided information, one school scores 65 points with no incorrect answers.

2. Understand the scoring details

From the question, we gather that:

• School A has a score of 65 for answering questions correctly.
• The scoring system utilizes terms such as $12x$, $5x$, and constants like $165$ and $65$.

3. Set up equations based on the given conditions

We need to create an equation that represents the scoring situation. Each of the options involves rearranging these scoring points into an equation format.

4. Examine the answer options

Let’s analyze each option:

• Option A: $12x + 5x - 165 = -12x + 65$

• Option B: $12x - 5x + 165 = 12x + 65$

• Option C: $5x - 12x + 165 = 12x + 65$

• Option D: $12x - 5x - 165 = 12x + 65$

5. Simplify each equation

We need to simplify each option to see which one logically represents the scoring situation correctly:

• For Option A: Combine like terms and rearrange to isolate 0 on one side. $$ 17x - 165 = -12x + 65 \implies 29x = 230 \implies x = \frac{230}{29} $$

• For Option B: $$ 12x - 5x + 165 = 12x + 65 \implies 7x + 165 = 12x + 65 \implies -5x = -100 \implies x = 20 $$

• For Option C: $$ 5x - 12x + 165 = 12x + 65 \implies -7x + 165 = 12x + 65 \implies -19x = -100 \implies x = \frac{100}{19} $$

• For Option D: $$ 12x - 5x - 165 = 12x + 65 \implies 7x - 165 = 12x + 65 \implies -5x = 230 \implies x = -46 $$

6. Determine the logical option

After evaluating, we compare each solution’s values, focusing particularly on the one critical that meets scoring logic and given constraints.