Solve the equation for x, y, z and t, if `2[x, z , y, t]+3[1, -1 , 0 , 2]=3[3, 5, 4, 6]`

Explanation

To solve the equation \( 2[x, z, y, t] + 3[1, -1, 0, 2] = 3[3, 5, 4, 6] \), we will follow these steps:

### Step 1: Write the equation in matrix form
We start with the equation:
\[ 2[x, z, y, t] + 3[1, -1, 0, 2] = 3[3, 5, 4, 6] \]

### Step 2: Multiply the matrices by their respective scalars
Multiply the first matrix by 2:
\[ 2[x, z, y, t] = [2x, 2z, 2y, 2t] \]
Multiply the second matrix by 3:
\[ 3[1, -1, 0, 2] = [3 \cdot 1, 3 \cdot -1, 3 \cdot 0, 3 \cdot 2] = [3, -3, 0, 6] \]
Multiply the third matrix by 3:
\[ 3[3, 5, 4, 6] = [3 \cdot 3, 3 \cdot 5, 3 \cdot 4, 3 \cdot 6] = [9, 15, 12, 18] \]

### Step 3: Substitute the results back into the equation
Now we have:
\[ [2x, 2z, 2y, 2t] + [3, -3, 0, 6] = [9, 15, 12, 18] \]

### Step 4: Combine the matrices
Add the two matrices on the left:
\[ [2x + 3, 2z - 3, 2y + 0, 2t + 6] = [9, 15, 12, 18] \]

### Step 5: Set up equations by comparing corresponding elements
From the above equation, we can set up the following equations:
1. \( 2x + 3 = 9 \)
2. \( 2z - 3 = 15 \)
3. \( 2y = 12 \)
4. \( 2t + 6 = 18 \)

### Step 6: Solve each equation for \( x, y, z, t \)

1. **Solving for \( x \)**:
\[ 2x + 3 = 9 \\ 2x = 9 - 3 \\ 2x = 6 \\ x = \frac{6}{2} = 3 \]

2. **Solving for \( z \)**:
\[ 2z - 3 = 15 \\ 2z = 15 + 3 \\ 2z = 18 \\ z = \frac{18}{2} = 9 \]

3. **Solving for \( y \)**:
\[ 2y = 12 \\ y = \frac{12}{2} = 6 \]

4. **Solving for \( t \)**:
\[ 2t + 6 = 18 \\ 2t = 18 - 6 \\ 2t = 12 \\ t = \frac{12}{2} = 6 \]

### Step 7: Write the final solution
The values of \( x, y, z, \) and \( t \) are:
\[ x = 3, \quad y = 6, \quad z = 9, \quad t = 6 \]

### Summary of the solution:
The final answer is:
\[ (x, y, z, t) = (3, 6, 9, 6) \]

---