If y=0 then the number of values of the pair (x, y) such that `x+y+x/y=1/2` and `(x+y)x/y= -1/2` is:

Options

A. 1
B. 2
C. 0
D. None of these

Answer

To solve the problem, we need to analyze the two equations given and find the values of the pair (x, y) when y = 0.

### Step 1: Substitute y = 0 into the equations
We start with the two equations:
1. \( x + y + \frac{x}{y} = \frac{1}{2} \)
2. \( (x + y) \frac{x}{y} = -\frac{1}{2} \)

Substituting \( y = 0 \) into these equations will lead to undefined expressions because division by zero occurs. Thus, we need to analyze the equations without substituting y = 0 directly.

### Step 2: Analyze the first equation
The first equation becomes:
\[ x + 0 + \frac{x}{0} = \frac{1}{2} \]
This is undefined due to the term \( \frac{x}{0} \).

### Step 3: Analyze the second equation
The second equation becomes:
\[ (x + 0) \frac{x}{0} = -\frac{1}{2} \]
This is also undefined due to the term \( \frac{x}{0} \).

### Step 4: Conclusion
Since both equations become undefined when \( y = 0 \), there are no valid pairs (x, y) that satisfy the equations under this condition.

### Final Answer
Thus, the number of values of the pair (x, y) such that the equations hold true when \( y = 0 \) is **0**.

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