To solve the equation \( |x - 4| + |y - 4| = 4 \) and find the number of integer values that the set \((x, y)\) can have, we can follow these steps:
### Step 1: Understand the Equation
The equation \( |x - 4| + |y - 4| = 4 \) represents a diamond (or rhombus) shape on the coordinate plane, centered at the point (4, 4). The distance from the center to the vertices of the diamond is 4.
### Step 2: Identify the Vertices
To find the vertices of the diamond, we can set \( |x - 4| \) and \( |y - 4| \) to different combinations that sum to 4. The vertices can be calculated as follows:
- When \( x - 4 = 4 \) and \( y - 4 = 0 \) → \( (8, 4) \)
- When \( x - 4 = 0 \) and \( y - 4 = 4 \) → \( (4, 8) \)
- When \( x - 4 = -4 \) and \( y - 4 = 0 \) → \( (0, 4) \)
- When \( x - 4 = 0 \) and \( y - 4 = -4 \) → \( (4, 0) \)
Thus, the vertices of the diamond are \( (8, 4) \), \( (4, 8) \), \( (0, 4) \), and \( (4, 0) \).
### Step 3: Determine the Boundary Lines
The equation can be broken down into four linear equations based on the cases of the absolute values:
1. \( x - 4 + y - 4 = 4 \) → \( x + y = 12 \)
2. \( x - 4 - (y - 4) = 4 \) → \( x - y = 8 \)
3. \( - (x - 4) + (y - 4) = 4 \) → \( -x + y = 0 \) → \( y = x \)
4. \( - (x - 4) - (y - 4) = 4 \) → \( -x - y = -8 \) → \( x + y = -4 \)
### Step 4: Find Integer Solutions
To find integer solutions, we can look for integer points that lie on the lines defined by the equations above within the bounds of the diamond.
- For \( x + y = 12 \): Possible integer pairs are \( (8, 4), (7, 5), (6, 6), (5, 7), (4, 8) \) → 5 points.
- For \( x - y = 8 \): Possible integer pairs are \( (8, 0), (9, 1), (10, 2), (11, 3), (12, 4) \) → 5 points.
- For \( y = x \): Possible integer pairs are \( (0, 0), (1, 1), (2, 2), (3, 3), (4, 4) \) → 5 points.
- For \( x + y = -4 \): This line does not intersect the diamond as it lies outside the bounds.
### Step 5: Count Unique Integer Points
Now we will count all unique integer points obtained from the lines:
- From \( x + y = 12 \): 5 points
- From \( x - y = 8 \): 5 points
- From \( y = x \): 5 points
However, we need to ensure that we do not double count any points. The unique integer points that satisfy the original equation are:
- \( (8, 4), (7, 5), (6, 6), (5, 7), (4, 8) \)
- \( (8, 0), (9, 1), (10, 2), (11, 3), (12, 4) \)
- \( (0, 0), (1, 1), (2, 2), (3, 3), (4, 4) \)
After careful counting, we find that there are a total of **16 unique integer points**.
### Final Answer
Thus, the number of integer values that the set \( (x, y) \) can have is **16**.
---
