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a. Write a system of equations to represent each student's comment. Be sure to specify what your variables represent.

Let [tex]\( E \)[/tex] be the points Elena scored.
Let [tex]\( K \)[/tex] be the points Kiran scored.

From Elena's comment:
[tex]\[ K = 2E \][/tex]

From Kiran's comment:
[tex]\[ K = E + 9 \][/tex]

b. If both students were correct, how many points did each student score? Show your reasoning.

Solve the system of equations:
[tex]\[ K = 2E \][/tex]
[tex]\[ K = E + 9 \][/tex]

Substitute the first equation into the second:
[tex]\[ 2E = E + 9 \][/tex]
[tex]\[ E = 9 \][/tex]

Substitute [tex]\( E \)[/tex] back into the first equation:
[tex]\[ K = 2 \times 9 \][/tex]
[tex]\[ K = 18 \][/tex]

Elena scored 9 points and Kiran scored 18 points.

Sagot :

GinnyAnswer
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.
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