If `[{:(,x-y,1,z),(,2x-y,0,w):}]=[{:(,-1,1,4),(,0,0,5):}], "find x,y,z,w"`

Answer

To solve the given problem, we need to equate the corresponding elements of the two matrices and solve for the variables \( x, y, z, \) and \( w \).

### Step-by-Step Solution:

1. **Write down the given matrices:**
\[ \begin{pmatrix} x - y & 1 & z \\ 2x - y & 0 & w \end{pmatrix} = \begin{pmatrix} -1 & 1 & 4 \\ 0 & 0 & 5 \end{pmatrix} \]

2. **Equate the corresponding elements:**
- From the first row, first column:
\[ x - y = -1 \quad \text{(1)} \]
- From the first row, second column:
\[ 1 = 1 \quad \text{(True, no new information)} \]
- From the first row, third column:
\[ z = 4 \quad \text{(2)} \]
- From the second row, first column:
\[ 2x - y = 0 \quad \text{(3)} \]
- From the second row, second column:
\[ 0 = 0 \quad \text{(True, no new information)} \]
- From the second row, third column:
\[ w = 5 \quad \text{(4)} \]

3. **Substitute the values of \( z \) and \( w \):**
- From (2), we have \( z = 4 \).
- From (4), we have \( w = 5 \).

4. **Now solve for \( x \) and \( y \):**
- We have two equations (1) and (3):
\[ x - y = -1 \quad \text{(1)} \]
\[ 2x - y = 0 \quad \text{(3)} \]

5. **Subtract equation (1) from equation (3):**
\[ (2x - y) - (x - y) = 0 - (-1) \]
Simplifying this gives:
\[ 2x - y - x + y = 1 \]
\[ x = 1 \quad \text{(5)} \]

6. **Substitute the value of \( x \) back into equation (1) to find \( y \):**
\[ 1 - y = -1 \]
Rearranging gives:
\[ y = 2 \quad \text{(6)} \]

7. **Final values:**
- From (5), \( x = 1 \)
- From (6), \( y = 2 \)
- From (2), \( z = 4 \)
- From (4), \( w = 5 \)

Thus, the final answers are:
\[ x = 1, \quad y = 2, \quad z = 4, \quad w = 5 \]