For any real numbers r and s such that `r nes`, let `r…s` be defined by `r….s=(r-s)/(r+s)`. If `r-s=63 and r…s=9`, what is the value of r?

Answer

To solve the problem, we need to find the value of \( r \) given the equations \( r - s = 63 \) and \( r \ldots s = 9 \), where \( r \ldots s \) is defined as \( \frac{r - s}{r + s} \).

### Step-by-Step Solution:

1. **Write down the definitions and given equations:**
- We know that \( r \ldots s = \frac{r - s}{r + s} \).
- We are given \( r - s = 63 \) and \( r \ldots s = 9 \).

2. **Substitute the value of \( r \ldots s \) into the equation:**
- Substitute \( r \ldots s = 9 \) into the definition:
\[ \frac{r - s}{r + s} = 9 \]

3. **Substitute \( r - s = 63 \) into the equation:**
- Replace \( r - s \) with 63:
\[ \frac{63}{r + s} = 9 \]

4. **Cross-multiply to eliminate the fraction:**
- Multiply both sides by \( r + s \):
\[ 63 = 9(r + s) \]

5. **Solve for \( r + s \):**
- Divide both sides by 9:
\[ r + s = \frac{63}{9} = 7 \]

6. **Now we have two equations:**
- \( r - s = 63 \) (Equation 1)
- \( r + s = 7 \) (Equation 2)

7. **Add the two equations:**
- Adding Equation 1 and Equation 2:
\[ (r - s) + (r + s) = 63 + 7 \]
- This simplifies to:
\[ 2r = 70 \]

8. **Solve for \( r \):**
- Divide both sides by 2:
\[ r = \frac{70}{2} = 35 \]

### Final Answer:
The value of \( r \) is \( 35 \).