Bryce took a quiz that consisted of true-or-false questions and multiple-choice questions. The true-or-false questions were worth 1 point each, and the multiple-choice questions were worth 2 points each. Bryce only knows that he answered 8 questions correctly and earned a total of 14 points. To determine the number of each type of question that he answered correctly, he wrote the system of linear equations x + y = 8 and x + 2 y = 14.

Question

Which ordered pair is a solution to this system of linear equations, and what does the ordered pair represent?
(0, 8); Bryce answered 0 true-or-false questions correctly and 8 multiple-choice questions correctly.
(2, 6); Bryce answered 2 true-or-false questions correctly and 6 multiple-choice questions correctly.
(6, 2); Bryce answered 6 true-or-false questions correctly and 2 multiple-choice questions correctly.
(8, 0); Bryce answered 8 true-or-false questions correctly and 0 multiple-choice questions correctly.

Answer 1

To solve for the ordered pair that is a solution to the system of equations, we can check each option against the equations:

1. First equation: \( x + y = 8 \) (where \( x \) is the number of true-or-false questions answered correctly, and \( y \) is the number of multiple-choice questions answered correctly).
2. Second equation: \( x + 2y = 14 \).

We will evaluate each of the given ordered pairs:

1. (0, 8):

   • \( x + y = 0 + 8 = 8 \) (True)
   • \( x + 2y = 0 + 2(8) = 16 \) (False)
   • This does not satisfy the second equation.

2. (2, 6):

   • \( x + y = 2 + 6 = 8 \) (True)
   • \( x + 2y = 2 + 2(6) = 2 + 12 = 14 \) (True)
   • This satisfies both equations.

3. (6, 2):

   • \( x + y = 6 + 2 = 8 \) (True)
   • \( x + 2y = 6 + 2(2) = 6 + 4 = 10 \) (False)
   • This does not satisfy the second equation.

4. (8, 0):

   • \( x + y = 8 + 0 = 8 \) (True)
   • \( x + 2y = 8 + 2(0) = 8 + 0 = 8 \) (False)
   • This does not satisfy the second equation.

The only pair that satisfies both equations is (2, 6).

This ordered pair represents: Bryce answered 2 true-or-false questions correctly and 6 multiple-choice questions correctly.