To solve the equations involving \(x\) and \(y\) given in the image, we will use the conditions and information provided:

1. **Equations to Solve:**
- \(xy = 5\) and \(x > y\)
- \(xy = 7\)
- \((x + 1)(y + 2) = 5\)
- \((x + 1)(y + 3) = 6\)
- \((x + 2)(y + 3) = 6\)
- \((x - 1)(y + 3) = 6\)
- \((x - 1)(y - 3) = 5\)
- \((x - 2)(y - 1) = 3\)
- \((x - 2)(y + 1) = 5\)
- \((x - 3)(y + 1) = 7\)

2. **Finding \(x\) and \(y\):**
- For the first equation \(xy = 5\) and \(x > y\), pairs of natural numbers that satisfy this are \((5, 1)\).
- For the second equation \(xy = 7\), possible pairs are \((7, 1)\) and \((3, 2)\) (but \(x\) must be greater than \(y\)).

3. **Check Additional Conditions:**
- Plugging the values of \(x\) and \(y\) into the other equations will help us find specific pairs.
- Start with pairs satisfying the first two conditions and progressively check which combinations solve the additional equations.

4. **Continue Solving:**
Continue substituting possible \(x\) and \(y\) values into remaining equations to find compatible values.

If you want step-by-step calculations for a particular equation or further clarifications, let me know!