Which of the given values of x and y make the following pair of matrices equal `[(3x+2,5),(y+1,2-3x)],[(0,y-2),(8,4)]`:

Options

A. `x=(-1)/(3),y=7`
B. Not possible to find
C. `y=7,x=(-2)/(3)`
D. `x=(-1)/(3),y=(-2)/(3)`.

Answer

To determine the values of \( x \) and \( y \) that make the following pair of matrices equal:

\[ \begin{pmatrix} 3x + 2 & 5 \\ y + 1 & 2 - 3x \end{pmatrix} = \begin{pmatrix} 0 & y - 2 \\ 8 & 4 \end{pmatrix} \]

we need to equate the corresponding elements of the matrices.

### Step 1: Equate the corresponding elements

1. **First Element**:
\[ 3x + 2 = 0 \]

2. **Second Element**:
\[ 5 = y - 2 \]

3. **Third Element**:
\[ y + 1 = 8 \]

4. **Fourth Element**:
\[ 2 - 3x = 4 \]

### Step 2: Solve for \( x \)

From the first equation:
\[ 3x + 2 = 0 \]
Subtract 2 from both sides:
\[ 3x = -2 \]
Now divide by 3:
\[ x = -\frac{2}{3} \]

### Step 3: Solve for \( y \)

From the second equation:
\[ 5 = y - 2 \]
Add 2 to both sides:
\[ y = 5 + 2 = 7 \]

### Step 4: Verify with the third equation

From the third equation:
\[ y + 1 = 8 \]
Substituting \( y = 7 \):
\[ 7 + 1 = 8 \]
This is true.

### Step 5: Verify with the fourth equation

From the fourth equation:
\[ 2 - 3x = 4 \]
Substituting \( x = -\frac{2}{3} \):
\[ 2 - 3\left(-\frac{2}{3}\right) = 2 + 2 = 4 \]
This is also true.

### Conclusion

The values of \( x \) and \( y \) that make the matrices equal are:
\[ x = -\frac{2}{3}, \quad y = 7 \]