Consider the equation `x+y-[x][y]=0`, where `[*]` is the greatest integer function.
Equation of one of the lines on which the non-integral solution of given equation lies is:

Options

A. (a) `x+y=-1`
B. (b) `x+y=0`
C. (c) `x+y=1`
D. (d) `x+y=5`

Answer

To solve the equation \( x + y - [x][y] = 0 \), where \([*]\) denotes the greatest integer function (floor function), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation:
\[ x + y = [x][y] \]
This indicates that the sum of \(x\) and \(y\) equals the product of their greatest integer parts.

### Step 2: Express \(x\) and \(y\) in terms of their integer and fractional parts
Let:
\[ x = [x] + \{x\} \quad \text{and} \quad y = [y] + \{y\} \]
where \(\{x\}\) and \(\{y\}\) are the fractional parts of \(x\) and \(y\) respectively. Thus, we can rewrite the equation as:
\[ ([x] + \{x\}) + ([y] + \{y\}) = [x][y] \]
This simplifies to:
\[ [x] + [y] + \{x\} + \{y\} = [x][y] \]

### Step 3: Rearranging the equation
Rearranging gives:
\[ \{x\} + \{y\} = [x][y] - ([x] + [y]) \]
Let \(a = [x]\) and \(b = [y]\). The equation becomes:
\[ \{x\} + \{y\} = ab - (a + b) \]

### Step 4: Analyze the fractional parts
Since \(\{x\}\) and \(\{y\}\) are both in the range \(0 < \{x\}, \{y\} < 1\), their sum must satisfy:
\[ 0 < \{x\} + \{y\} < 2 \]
This implies:
\[ 0 < ab - (a + b) < 2 \]

### Step 5: Finding integer solutions
From the inequality \(ab - (a + b) > 0\), we can rewrite it as:
\[ ab > a + b \]
This can be factored as:
\[ (a - 1)(b - 1) > 1 \]
Now we need to find integer pairs \((a, b)\) that satisfy this condition.

### Step 6: Possible integer pairs
1. If \(a = 2\), then \(b\) must be \(3\) (since \(1 \cdot 2 > 1\)).
2. If \(a = 3\), then \(b\) must be \(2\) (same reasoning).
3. Negative cases can be checked as well:
- If \(a = 0\), \(b = -1\) or vice versa.

### Step 7: Checking the results
From the pairs:
- For \(a = 2, b = 3\) or \(a = 3, b = 2\):
\[ x + y = 2 \cdot 3 = 6 \]
- For \(a = 0, b = -1\) or vice versa:
\[ x + y = 0 \cdot -1 = 0 \]

### Conclusion
Thus, the possible equations for the lines on which the non-integral solutions lie are:
1. \(x + y = 6\)
2. \(x + y = 0\)

Among the options provided, the equation of one of the lines is:
\[ \boxed{x + y = 0} \]