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.

Question

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.

Answer 1

To solve the system of equations given:

1. \(x + y = 24\) (equation 1)
2. \(3x + 5y = 100\) (equation 2)

We can use substitution or elimination. Let's use substitution. From equation 1, we can express \(y\) in terms of \(x\):

\[ y = 24 - x \]

Now, we can substitute this expression for \(y\) into equation 2:

\[ 3x + 5(24 - x) = 100 \]

Distributing \(5\) gives:

\[ 3x + 120 - 5x = 100 \]

Combining like terms:

\[ -2x + 120 = 100 \]

To isolate \(x\), subtract \(120\) from both sides:

\[ -2x = -20 \]

Dividing by \(-2\) gives:

\[ x = 10 \]

Now that we have \(x\), we can find \(y\) using equation 1:

\[ y = 24 - x = 24 - 10 = 14 \]

So, we have:

\[ x = 10 \quad \text{(3-point questions)} \] \[ y = 14 \quad \text{(5-point questions)} \]

Thus, the test contains 10 three-point questions and 14 five-point questions.

The correct answer is:

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