` ( 5 ) /( x+ 1 ) - (2)/(y - 1 ) = (1)/(2)`,
` (10)/(x + 1) + ( 2 ) /( y - 1) = ( 5)/(2), x ne - 1 and y ne 1 `

Explanation

To solve the given system of equations:

1. **Equation 1**: \(\frac{5}{x+1} - \frac{2}{y-1} = \frac{1}{2}\)
2. **Equation 2**: \(\frac{10}{x+1} + \frac{2}{y-1} = \frac{5}{2}\)

We will use substitution to simplify the equations.

### Step 1: Substitute Variables
Let:
- \( u = \frac{5}{x+1} \)
- \( v = \frac{2}{y-1} \)

Now, we can rewrite the equations as:
- **Equation 1**: \( u - v = \frac{1}{2} \) (1)
- **Equation 2**: \( 2u + v = \frac{5}{2} \) (2)

### Step 2: Solve for \( u \) and \( v \)
Now, we will solve these two equations simultaneously.

From Equation (1):
\[ v = u - \frac{1}{2} \]

Substituting \( v \) in Equation (2):
\[ 2u + (u - \frac{1}{2}) = \frac{5}{2} \]
Combine like terms:
\[ 3u - \frac{1}{2} = \frac{5}{2} \]

### Step 3: Isolate \( u \)
Add \(\frac{1}{2}\) to both sides:
\[ 3u = \frac{5}{2} + \frac{1}{2} \]
\[ 3u = \frac{6}{2} \]
\[ 3u = 3 \]
Now divide by 3:
\[ u = 1 \]

### Step 4: Find \( v \)
Substituting \( u = 1 \) back into Equation (1):
\[ 1 - v = \frac{1}{2} \]
\[ v = 1 - \frac{1}{2} \]
\[ v = \frac{1}{2} \]

### Step 5: Substitute Back to Find \( x \) and \( y \)
Now we have:
- \( u = 1 \) implies \( \frac{5}{x+1} = 1 \)
- \( v = \frac{1}{2} \) implies \( \frac{2}{y-1} = \frac{1}{2} \)

From \( \frac{5}{x+1} = 1 \):
\[ 5 = x + 1 \]
\[ x = 5 - 1 \]
\[ x = 4 \]

From \( \frac{2}{y-1} = \frac{1}{2} \):
Cross-multiply:
\[ 2 = \frac{1}{2}(y - 1) \]
Multiply both sides by 2:
\[ 4 = y - 1 \]
\[ y = 4 + 1 \]
\[ y = 5 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ (x, y) = (4, 5) \]