Given `3[x y z w]-[x6-1 2w]+[4x+y z+w3]` , find the values of x, y, z and w.

Explanation

To solve the equation \( 3[x \, y \, z \, w] = [x \, 6 \, -1 \, 2w] + [4x \, +y \, z \, +w \, 3] \), we will follow these steps:

### Step 1: Write the equation in matrix form
We can express the equation as:
\[ 3 \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} = \begin{bmatrix} x \\ 6 \\ -1 \\ 2w \end{bmatrix} + \begin{bmatrix} 4x \\ y \\ z \\ w \\ 3 \end{bmatrix} \]

### Step 2: Combine the right-hand side
Now, we will add the two matrices on the right-hand side:
\[ \begin{bmatrix} x + 4x \\ 6 + y \\ -1 + z \\ 2w + 3 \end{bmatrix} = \begin{bmatrix} 5x \\ 6 + y \\ -1 + z \\ 2w + 3 \end{bmatrix} \]

### Step 3: Set the left-hand side equal to the right-hand side
Now we have:
\[ 3 \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} = \begin{bmatrix} 5x \\ 6 + y \\ -1 + z \\ 2w + 3 \end{bmatrix} \]

### Step 4: Write the equations for each element
From the equality of the matrices, we can write the following equations:
1. \( 3x = 5x \)
2. \( 3y = 6 + y \)
3. \( 3z = -1 + z \)
4. \( 3w = 2w + 3 \)

### Step 5: Solve each equation

**Equation 1:**
\[ 3x = 5x \implies 3x - 5x = 0 \implies -2x = 0 \implies x = 0 \]

**Equation 2:**
\[ 3y = 6 + y \implies 3y - y = 6 \implies 2y = 6 \implies y = 3 \]

**Equation 3:**
\[ 3z = -1 + z \implies 3z - z = -1 \implies 2z = -1 \implies z = -\frac{1}{2} \]

**Equation 4:**
\[ 3w = 2w + 3 \implies 3w - 2w = 3 \implies w = 3 \]

### Step 6: Summary of solutions
The values are:
- \( x = 0 \)
- \( y = 3 \)
- \( z = -\frac{1}{2} \)
- \( w = 3 \)