To solve this problem, we will first set up the expressions for the scores and then derive the total points scored in the game.
1. Define Central's score before the three-pointer:
Let [tex]\( c \)[/tex] be Central High School's score before they scored the three-pointer.
2. Eastern High School's score:
Eastern had double the score of Central before the three-pointer.
Therefore, Eastern's score is [tex]\( 2c \)[/tex].
3. Central's score after scoring the three-pointer:
Central’s score after the three-pointer is [tex]\( c + 3 \)[/tex].
4. Total points scored in the game:
The total points scored in the game is the sum of Eastern’s score and Central’s final score.
So, the total points is [tex]\( 2c + (c + 3) \)[/tex].
Now, we simplify this expression step-by-step:
[tex]\[
2c + (c + 3) = 2c + c + 3 = 3c + 3
\][/tex]
With this expression [tex]\( 3c + 3 \)[/tex] in mind, let’s verify each of the given options:
1. [tex]\( 2c + c \)[/tex]:
[tex]\[
2c + c = 3c
\][/tex]
2. [tex]\( 3c + 3 \)[/tex]:
This expression matches our simplified result:
[tex]\[
3c + 3
\][/tex]
3. [tex]\( 2c + 3 \)[/tex]:
This expression does not correspond directly to our simplified result.
4. [tex]\( 2c + c - 3 \)[/tex]:
[tex]\[
2c + c - 3 = 3c - 3
\][/tex]
5. [tex]\( 2c - c + 3 \)[/tex]:
[tex]\[
2c - c + 3 = c + 3
\][/tex]
6. [tex]\( 2c + c + 3 \)[/tex]:
This expression corresponds directly to our simplified result:
[tex]\[
2c + c + 3 = 3c + 3
\][/tex]
Therefore, the options that match the total points scored in the game correctly are:
- [tex]\( 2c + c \)[/tex]
- [tex]\( 3c + 3 \)[/tex]
- [tex]\( 2c + c + 3 \)[/tex]
So the checked expressions are [tex]\(2c+c, 3c+3\)[/tex], and [tex]\(2c+c+3\)[/tex].
