Select the correct answer.
On a television game show, contestants earn 5 points for each correct answer, but lose 2 points for each wrong answer. On last week's show, the winning contestant earned 80 points and answered 30 questions.
Given that [tex]$x$[/tex] represents the number of questions answered correctly, and [tex]$y$[/tex] represents the number of questions answered incorrectly, determine which system of equations correctly represents this situation.
A. [tex]$\begin{aligned}
5 x-2 y & =30 \\
x+y & =80
\end{aligned}$[/tex]
B. [tex]$\begin{array}{r}
5 x-2 y=80 \\
x+y=30
\end{array}$[/tex]
C. [tex]$\begin{aligned}
5 x+2 y & =30 \\
x-y & =80
\end{aligned}$[/tex]
D. [tex]$\begin{array}{r}
5 x+2 y=80 \\
x-y=30
\end{array}$[/tex]
Answer (2)
Set up an equation for the total points earned: 5 x − 2 y = 80 .
Set up an equation for the total number of questions answered: x + y = 30 .
Combine the two equations to form a system of equations.
The correct system of equations is { 5 x − 2 y = 80 x + y = 30 , so the final answer is { 5 x − 2 y = 80 x + y = 30 .
Explanation
Problem Analysis Let's analyze the problem. We are given that contestants earn 5 points for each correct answer and lose 2 points for each wrong answer. The winning contestant earned 80 points and answered 30 questions. We are also given that x represents the number of questions answered correctly, and y represents the number of questions answered incorrectly.
Setting up Equations We can set up two equations based on the given information. The first equation represents the total points earned, which is 80. Since each correct answer earns 5 points and each wrong answer loses 2 points, the equation is: 5 x − 2 y = 80 The second equation represents the total number of questions answered, which is 30. Since x is the number of correct answers and y is the number of incorrect answers, the equation is: x + y = 30
System of Equations Combining these two equations, we get the following system of equations: { 5 x − 2 y = 80 x + y = 30
Finding the Correct Option Now, we compare this system of equations with the given options. The correct system of equations is: { 5 x − 2 y = 80 x + y = 30
Final Answer Therefore, the correct answer is: { 5 x − 2 y = 80 x + y = 30
Examples
Imagine you're organizing a school quiz where participants score points for correct answers and lose points for incorrect ones. This problem helps you determine the number of correct and incorrect answers a participant gave, based on their total score and the total number of questions they answered. Understanding how to set up and solve such systems of equations can be applied to various scenarios, such as calculating grades, managing budgets, or analyzing game statistics. This algebraic approach provides a structured way to analyze and interpret real-world situations involving multiple variables and constraints.
The correct system of equations representing the game show scenario is found in Option B: { 5 x − 2 y = 80 x + y = 30 . This system comes from how points are calculated for correct and incorrect answers. The contestant earned 80 points and answered a total of 30 questions.
;
