A science test, which is worth 100 points, consists of 24 questions. Each question is worth either 3 points or 5 points. If [tex]$x$[/tex] is the number of 3-point questions and [tex]$y$[/tex] is the number of 5-point questions, the system shown represents this situation.
[tex]
\begin{array}{c}
x+y=24 \\
3 x+5 y=100
\end{array}
[/tex]
What does the solution of this system indicate about the questions on the test?
A. The test contains 4 three-point questions and 20 five-point questions.
B. The test contains 10 three-point questions and 14 five-point questions.
C. The test contains 14 three-point questions and 10 five-point questions.
D. The test contains 20 three-point questions and 8 five-point questions.
Answer (1)
Express x in terms of y using the first equation: x = 24 − y .
Substitute this expression into the second equation and solve for y : 3 ( 24 − y ) + 5 y = 100 ⇒ y = 14 .
Substitute the value of y back into the expression for x : x = 24 − 14 ⇒ x = 10 .
The solution indicates that the test contains 10 three-point questions and 14 five-point questions: 10 three-point questions and 14 five-point questions .
Explanation
Analyze the problem We are given a system of two equations with two variables, x and y , representing the number of 3-point and 5-point questions on a science test, respectively. The system is:
{ x + y = 24 3 x + 5 y = 100
Our goal is to solve this system to find the values of x and y , which will tell us how many 3-point and 5-point questions are on the test.
Solve for y We can solve this system using substitution or elimination. Let's use substitution. From the first equation, we can express x in terms of y :
x = 24 − y
Now, substitute this expression for x into the second equation:
3 ( 24 − y ) + 5 y = 100
Expand and simplify:
72 − 3 y + 5 y = 100
2 y = 100 − 72
2 y = 28
y = 14
So, there are 14 five-point questions.
Solve for x Now that we have the value of y , we can find the value of x using the expression x = 24 − y :
x = 24 − 14
x = 10
So, there are 10 three-point questions.
State the solution The solution to the system is x = 10 and y = 14 . This means there are 10 three-point questions and 14 five-point questions on the test.
Examples
Understanding systems of equations is crucial in many real-world scenarios. For instance, imagine you're managing a store selling two types of products. If you know the total number of products sold and the total revenue, you can set up a system of equations to determine how many of each product were sold. This helps in inventory management and understanding customer preferences, allowing you to make informed decisions about restocking and marketing strategies. This same approach can be applied to resource allocation, mixture problems, and various other practical situations.
