Let's determine the correct system of equations based on the information given in the problem.
Let's denote:
- [tex]\( x \)[/tex] as the number of questions answered correctly.
- [tex]\( y \)[/tex] as the number of questions answered incorrectly.
We need to set up two equations based on the problem's conditions:
1. Points Condition:
- Each correct answer gives 5 points.
- Each incorrect answer deducts 2 points.
- The total points earned by the contestant is 80.
So, the equation reflecting the points condition is:
[tex]\[
5x - 2y = 80
\][/tex]
2. Total Questions Condition:
- The contestant answered a total of 30 questions.
So, the equation reflecting the total number of questions is:
[tex]\[
x + y = 30
\][/tex]
Thus, the system of equations that correctly represents the situation is:
[tex]\[
\begin{array}{r}
5x - 2y = 80 \\
x + y = 30
\end{array}
\][/tex]
Notice that this matches the following system provided in the options:
[tex]\[
\begin{array}{r}
5x - 2y = 80 \\
x + y = 30
\end{array}
\][/tex]
So, the correct system of equations is:
[tex]\[
\begin{array}{r}
5 x - 2 y = 80 \\
x + y = 30
\end{array}
\][/tex]
