Of course! Let's solve this problem step by step.
### a. Write a system of equations to represent each student's comment.
First, we need to define our variables:
- Let [tex]\( x \)[/tex] be the number of points Elena scored.
- Let [tex]\( y \)[/tex] be the number of points Kiran scored.
Now, according to Elena's comment: "You (Kiran) earned twice mine!"
This can be translated into the equation:
[tex]\[ y = 2x \][/tex]
According to Kiran's comment: "I only scored 9 points higher than you did."
This can be written as:
[tex]\[ y = x + 9 \][/tex]
So, the system of equations representing the comments is:
1. [tex]\( y = 2x \)[/tex]
2. [tex]\( y = x + 9 \)[/tex]
### b. If both students were correct, how many points did each student score? Show your reasoning.
We have the system of equations:
[tex]\[ y = 2x \][/tex]
[tex]\[ y = x + 9 \][/tex]
Since both equations equal [tex]\( y \)[/tex], we can set them equal to each other:
[tex]\[ 2x = x + 9 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ 2x - x = x + 9 - x \][/tex]
[tex]\[ x = 9 \][/tex]
So, Elena scored 9 points. Now, substitute [tex]\( x = 9 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. We'll use [tex]\( y = 2x \)[/tex]:
[tex]\[ y = 2(9) \][/tex]
[tex]\[ y = 18 \][/tex]
Thus, Kiran scored 18 points.
In conclusion:
- Elena scored 9 points.
- Kiran scored 18 points.
