Question

Find the number of integer pairs (x, y) that satisfy the following system of inequalities:
\[ \begin{cases} x \geq y \geq 3 \\ x + y \leq 14 \end{cases} \]

Hint:

When counting integer points defined by linear inequalities, fix one variable and find the range for the second. Summing the number of possibilities for each fixed value gives the total count.

Answer 1

Correct Answer: 25

Step 1: Understanding the Question:
The question asks for the number of pairs of integers (x, y) that simultaneously satisfy three conditions: \(x \geq y\), \(y \geq 3\), and \(x + y \leq 14\). This is a counting problem based on linear inequalities.
Step 2: Key Formula or Approach:
We can solve this problem by iterating through all possible integer values for one variable (say, y) and counting the corresponding number of valid integer values for the other variable (x). The number of integers in a range [a, b] is given by \(b - a + 1\).
Step 3: Detailed Explanation:
From the given inequalities, we can establish bounds for x and y.
1. We know \(y\) is an integer and \(y \geq 3\).
2. We have two constraints on \(x\): \(x \geq y\) and \(x \leq 14 - y\).
For integer solutions for \(x\) to exist, the lower bound for \(x\) must be less than or equal to its upper bound:
\[ y \leq 14 - y \] \[ 2y \leq 14 \] \[ y \leq 7 \] Combining this with \(y \geq 3\), we find that the possible integer values for y are 3, 4, 5, 6, and 7.
Now, we can count the number of possible integer values for \(x\) for each value of \(y\):
The number of integers for \(x\) is given by the formula: \((14 - y) - y + 1 = 15 - 2y\).
- For y = 3:
Number of x values = \(15 - 2(3) = 15 - 6 = 9\). (The values are x = 3, 4, ..., 11).
- For y = 4:
Number of x values = \(15 - 2(4) = 15 - 8 = 7\). (The values are x = 4, 5, ..., 10).
- For y = 5:
Number of x values = \(15 - 2(5) = 15 - 10 = 5\). (The values are x = 5, 6, ..., 9).
- For y = 6:
Number of x values = \(15 - 2(6) = 15 - 12 = 3\). (The values are x = 6, 7, 8).
- For y = 7:
Number of x values = \(15 - 2(7) = 15 - 14 = 1\). (The value is x = 7).
To find the total number of integer pairs (x, y), we sum the counts for each possible value of y:
Total pairs = \(9 + 7 + 5 + 3 + 1\).
This is the sum of an arithmetic progression.
Total pairs = 25.
Step 4: Final Answer:
The total number of integer pairs (x, y) satisfying the conditions is 25.