Question

a science test, which is worth 100 points, consists of 24 questions. each question is worth either 3 points or 5 points. if x is the number of 3 - point questions and y is the number of 5 - point questions, the system shown represents this situation.
$x + y = 24$
$3x + 5y = 100$
what does the solution of this system indicate about the questions on the test?

• the test contains 4 three - point questions and 20 five - point questions.
• the test contains 10 three - point questions and 14 five - point questions.
• the test contains 14 three - point questions and 10 five - point questions.
• the test contains 20 three - point questions and 8 five - point questions.

Explanation:

Step1: Solve the system of equations

We have the system:
\[

$$\begin{cases} x + y = 24 \\ 3x + 5y = 100 \end{cases}$$

\]
From the first equation \(x + y = 24\), we can express \(x\) as \(x = 24 - y\).

Step2: Substitute \(x = 24 - y\) into the second equation

Substitute \(x = 24 - y\) into \(3x + 5y = 100\):
\[

$$\begin{align*} 3(24 - y) + 5y &= 100 \\ 72 - 3y + 5y &= 100 \\ 72 + 2y &= 100 \\ 2y &= 100 - 72 \\ 2y &= 28 \\ y &= 14 \end{align*}$$

\]

Step3: Find the value of \(x\)

Substitute \(y = 14\) into \(x = 24 - y\):
\[
x = 24 - 14 = 10
\]

Wait, that's not matching the first option. Wait, maybe I made a mistake. Wait, let's check again. Wait, maybe I should use elimination.

Multiply the first equation \(x + y = 24\) by 3: \(3x + 3y = 72\)

Subtract this from the second equation \(3x + 5y = 100\):

\[

$$\begin{align*} (3x + 5y) - (3x + 3y) &= 100 - 72 \\ 3x + 5y - 3x - 3y &= 28 \\ 2y &= 28 \\ y &= 14 \end{align*}$$

\]

Then \(x = 24 - 14 = 10\). Wait, but the first option is 4 and 20. Wait, maybe I messed up the equations. Wait, let's check the first option: 4 three-point and 20 five-point. Let's check if it satisfies the equations.

For \(x + y = 24\): 4 + 20 = 24, which is correct.

For \(3x + 5y = 100\): 34 + 520 = 12 + 100 = 112, which is not 100. So that's wrong.

Wait, my solution: x=10, y=14. Let's check: 10 +14=24, correct. 310 +514=30 +70=100, correct. So the correct option is "The test contains 10 three-point questions and 14 five-point questions."

Wait, but the first option was 4 and 20, which we saw is wrong. So the correct answer is the second option? Wait, the options are:

1. The test contains 4 three-point questions and 20 five-point questions.

1. The test contains 10 three-point questions and 14 five-point questions.

1. The test contains 14 three-point questions and 10 five-point questions.

1. The test contains 20 three-point questions and 8 five-point questions.

Wait, my solution is x=10 (three-point), y=14 (five-point), which is the second option. Wait, but let's check again.

Wait, maybe I misread the equations. The problem says "each question is worth either 3 points or 5 points. If x is the number of 3-point questions and y is the number of 5-point questions". So the system is x + y =24 (total questions) and 3x +5y=100 (total points). So solving gives x=10, y=14. So the second option is correct.

Wait, but earlier when I thought x=10, y=14, that's correct. So the answer is the second option: "The test contains 10 three-point questions and 14 five-point questions."

Answer:

The test contains 10 three - point questions and 14 five - point questions. (Corresponding to the option: The test contains 10 three - point questions and 14 five - point questions.)