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 the system of equations given by \(x + y = 8\) and \(x + 2y = 14\), we can substitute the values from the ordered pairs to find which one is a solution.

1. (0, 8):

   • Plugging in: \(x + y = 0 + 8 = 8\) (True)
   • \(x + 2y = 0 + 2(8) = 0 + 16 = 16\) (False)

2. (2, 6):

   • Plugging in: \(x + y = 2 + 6 = 8\) (True)
   • \(x + 2y = 2 + 2(6) = 2 + 12 = 14\) (True)

3. (6, 2):

   • Plugging in: \(x + y = 6 + 2 = 8\) (True)
   • \(x + 2y = 6 + 2(2) = 6 + 4 = 10\) (False)

4. (8, 0):

   • Plugging in: \(x + y = 8 + 0 = 8\) (True)
   • \(x + 2y = 8 + 2(0) = 8 + 0 = 8\) (False)

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

This means:

• Bryce answered 2 true-or-false questions correctly (worth 1 point each, totaling 2 points).
• Bryce answered 6 multiple-choice questions correctly (worth 2 points each, totaling 12 points).

Thus, the ordered pair (2, 6) represents that Bryce answered 2 true-or-false questions correctly and 6 multiple-choice questions correctly.