`y gt x+r`
`y lt s -x`
If `x =y=1` is a solution to the system of inequalities above, which of the following ordered pairs could correspond to `(r,s)` ?

Options

A. ` (-1,1)`
B. `(-(1)/(2), 2)`
C. `(-(1)/(10), 3)`
D. `(3,-1)`

Explanation

To solve the problem, we need to analyze the given inequalities and the condition that \( x = y = 1 \) is a solution.

### Step 1: Substitute \( x \) and \( y \) into the inequalities
We have the inequalities:
1. \( y > x + r \)
2. \( y < s - x \)

Substituting \( x = 1 \) and \( y = 1 \) into these inequalities gives us:
1. \( 1 > 1 + r \)
2. \( 1 < s - 1 \)

### Step 2: Simplify the inequalities
From the first inequality:
\[ 1 > 1 + r \]
Subtracting 1 from both sides:
\[ 0 > r \quad \text{or} \quad r < 0 \]

From the second inequality:
\[ 1 < s - 1 \]
Adding 1 to both sides:
\[ 2 < s \quad \text{or} \quad s > 2 \]

### Step 3: Determine the conditions for \( r \) and \( s \)
We have derived two conditions:
1. \( r < 0 \)
2. \( s > 2 \)

### Step 4: Evaluate the options for \( (r, s) \)
Now we need to check which of the given ordered pairs satisfies these conditions:

1. **Option A: \( (-1, 1) \)**
- \( r = -1 \) (satisfies \( r < 0 \))
- \( s = 1 \) (does not satisfy \( s > 2 \))
- **Not a solution.**

2. **Option B: \( (-\frac{1}{2}, 2) \)**
- \( r = -\frac{1}{2} \) (satisfies \( r < 0 \))
- \( s = 2 \) (does not satisfy \( s > 2 \))
- **Not a solution.**

3. **Option C: \( (-\frac{1}{10}, 3) \)**
- \( r = -\frac{1}{10} \) (satisfies \( r < 0 \))
- \( s = 3 \) (satisfies \( s > 2 \))
- **This is a solution.**

4. **Option D: \( (3, -1) \)**
- \( r = 3 \) (does not satisfy \( r < 0 \))
- \( s = -1 \) (does not satisfy \( s > 2 \))
- **Not a solution.**

### Conclusion
The only ordered pair that satisfies both conditions is **Option C: \( (-\frac{1}{10}, 3) \)**.