To determine the total points scored in the game, let's consider the variables and conditions given in the problem.
1. Let [tex]\( c \)[/tex] represent Central High School's score before they scored a three-pointer.
2. Eastern High School's score was double Central High School's score before the three-pointer, so Eastern's score is [tex]\( 2c \)[/tex].
3. Central High School's final score after scoring a three-pointer is [tex]\( c + 3 \)[/tex].
Now let's interpret and check each given expression to see if it accurately represents the total points scored in the game, both teams combined.
1. [tex]\( 2c + c \)[/tex]
- This expression represents Eastern's score ([tex]\( 2c \)[/tex]) plus Central's initial score ([tex]\( c \)[/tex]).
- This does not account for the three-pointer.
- Result: [tex]\( 0 \)[/tex]
2. [tex]\( 3c + 3 \)[/tex]
- This expression represents Central's initial score ([tex]\( c \)[/tex]) doubled (Eastern’s score, [tex]\( 2c \)[/tex]), plus Central’s final score including the three-pointer.
- It correctly accounts for Central’s score increase with the three-pointer.
- Result: [tex]\( 3 \)[/tex]
3. [tex]\( 2c + 3 \)[/tex]
- This expression represents just Eastern’s score ([tex]\( 2c \)[/tex]) plus 3.
- It does not adequately represent the total points scored by both teams.
- Result: [tex]\( 3 \)[/tex]
4. [tex]\( 2c + c - 3 \)[/tex]
- This expression represents Eastern’s score ([tex]\( 2c \)[/tex]) plus Central’s initial score ([tex]\( c \)[/tex]), but then subtracts 3.
- It improperly removes the three-pointer.
- Result: [tex]\( -3 \)[/tex]
5. [tex]\( 2c - c + 3 \)[/tex]
- This simplifies to [tex]\( c + 3 \)[/tex], which only represents Central’s final score, excluding Eastern’s score.
- This does not reflect the total points from both teams.
- Result: [tex]\( 3 \)[/tex]
6. [tex]\( 2c + c + 3 \)[/tex]
- This represents Eastern’s score ([tex]\( 2c \)[/tex]) plus Central’s initial score ([tex]\( c \)[/tex]) plus the three-pointer.
- It correctly includes the three-pointer to Central’s final score.
- Result: [tex]\( 3 \)[/tex]
To summarize, using the given options, the expressions that correctly represent the total points scored by both teams in the game are:
- [tex]\( 3c + 3 \)[/tex]
- [tex]\( 2c + 3 \)[/tex]
- [tex]\( 2c + c + 3 \)[/tex]
Overall, by consistently analyzing each expression step-by-step, you can understand how each mathematically represents (or fails to represent) the game's final score scenario.
