Next week your math teacher is giving a chapter test that has 35 questions on it.  Some problems  are worth 2 points and some  are worth 4 points.  What is the combination you must get correct to score a perfect grade on the test.   Type in answer as an ordered pair.   ( x,y)

To solve this problem, let's define:

- \( x \) as the number of 2-point questions
- \( y \) as the number of 4-point questions

We are given the following information:

1. There are 35 questions in total:
   \[ x + y = 35 \]
2. The total score from these questions must equal the perfect score. Since some questions are worth 2 points and others are worth 4 points, the total score is given by:
   \[ 2x + 4y \]

To score a perfect grade, we need to achieve the maximum possible score. Since the maximum score is not explicitly stated, we'll assume that all questions must be answered correctly for full credit. Let's set up the equations accordingly.

First, solve the first equation for \( x \):
\[ x = 35 - y \]

Substitute this into the second equation for total points:
\[ 2(35 - y) + 4y = \text{Maximum score} \]

Given that the problem doesn't specify the total score, we will solve for \( x \) and \( y \) based on the number of questions and the requirement to answer them all correctly.

Simplify the equation:
\[ 70 - 2y + 4y = \text{Maximum score} \]
\[ 70 + 2y = \text{Maximum score} \]

Assuming the maximum score is achieved by answering all questions correctly:
\[ 70 + 2y = 140 \] (since \( 2 \times 35 = 70 \) and \( 4 \times 35 = 140 \))

Solve for \( y \):
\[ 2y = 70 \]
\[ y = 35 \]

Substitute \( y = 35 \) back into the equation for \( x \):
\[ x = 35 - 35 = 0 \]

So, the solution is \( (x, y) = (0, 35) \), meaning you must get 0 two-point questions and 35 four-point questions correct to achieve a perfect score.

**Answer:** \( (0, 35) \)

Do you want more details or have any questions about the solution? Here are 5 related questions you might find interesting:

1. What if the total points needed for a perfect score were specified as 100 points?
2. How would the solution change if there were more types of questions with different point values?
3. Can you form a general formula for problems of this nature with different numbers of questions and point values?
4. How does the number of 4-point questions affect the total score compared to the number of 2-point questions?
5. What strategies can be used to optimize scoring if not all questions can be answered correctly?

**Tip:** When solving word problems, always start by identifying and defining variables for the unknown quantities. Then, set up equations based on the information given and solve them systematically.

Explore how to calculate the combination of correct answers required to achieve a perfect score on a test with 35 questions of varying point values. Learn about equations and scoring systems used in mathematics. Suitable for Grades 7-9.

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