If `[x,y] [{:(,x),(,y):}]=[25] and [-x,y] [{:(,2x),(,y):}]=[-2,]2` find x and y if:
(i) `x, y in W` (whole numbers)
(ii) `x,y in Z` (integers)

Answer

To solve the problem, we need to derive two equations from the given matrix equations and then find the values of \( x \) and \( y \) under the specified conditions.

### Step-by-Step Solution:

1. **Set Up the Equations:**
From the first matrix equation:
\[ [x, y] \cdot \begin{bmatrix} x \\ y \end{bmatrix} = 25 \]
This translates to:
\[ x^2 + y^2 = 25 \quad \text{(Equation 1)} \]

From the second matrix equation:
\[ [-x, y] \cdot \begin{bmatrix} 2x \\ y \end{bmatrix} = -2 \]
This translates to:
\[ -2x^2 + y^2 = -2 \quad \text{(Equation 2)} \]

2. **Rearranging Equation 2:**
Rearranging Equation 2 gives:
\[ y^2 = 2x^2 - 2 \quad \text{(Equation 3)} \]

3. **Substituting Equation 3 into Equation 1:**
Substitute Equation 3 into Equation 1:
\[ x^2 + (2x^2 - 2) = 25 \]
Simplifying this:
\[ x^2 + 2x^2 - 2 = 25 \]
\[ 3x^2 - 2 = 25 \]
\[ 3x^2 = 27 \]
\[ x^2 = 9 \]
Therefore:
\[ x = 3 \quad \text{or} \quad x = -3 \]

4. **Finding y using x:**
Substitute \( x = 3 \) into Equation 3:
\[ y^2 = 2(3^2) - 2 = 2(9) - 2 = 18 - 2 = 16 \]
Thus:
\[ y = 4 \quad \text{or} \quad y = -4 \]

5. **Summary of Solutions:**
- If \( x \) and \( y \) are whole numbers (W), the solutions are:
\[ (x, y) = (3, 4) \]
- If \( x \) and \( y \) are integers (Z), the solutions are:
\[ (x, y) = (3, 4), (-3, 4), (3, -4), (-3, -4) \]