Which of the given values of x and y make the following pair of matrices equal`[[3x+7, 5 ],[y+1, 2-3x]],[[0, y-2],[ 8, 4]]`

Options

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

Explanation

To solve the problem of finding the values of \( x \) and \( y \) that make the two matrices equal, we start by equating the corresponding elements of the matrices.

The two matrices given are:
\[ \begin{pmatrix} 3x + 7 & 5 \\ y + 1 & 2 - 3x \end{pmatrix} \]
and
\[ \begin{pmatrix} 0 & y - 2 \\ 8 & 4 \end{pmatrix} \]

### Step 1: Equate the corresponding elements of the matrices

1. **First Element**:
\[ 3x + 7 = 0 \]
**Second Element**:
\[ 5 = y - 2 \]
**Third Element**:
\[ y + 1 = 8 \]
**Fourth Element**:
\[ 2 - 3x = 4 \]

### Step 2: Solve the equations

**From the first equation**:
\[ 3x + 7 = 0 \implies 3x = -7 \implies x = -\frac{7}{3} \]

**From the second equation**:
\[ 5 = y - 2 \implies y = 5 + 2 = 7 \]

**From the third equation**:
\[ y + 1 = 8 \implies y = 8 - 1 = 7 \]

**From the fourth equation**:
\[ 2 - 3x = 4 \implies -3x = 4 - 2 \implies -3x = 2 \implies x = -\frac{2}{3} \]

### Step 3: Analyze the results

Now we have two different values for \( x \):
1. From the first equation: \( x = -\frac{7}{3} \)
2. From the fourth equation: \( x = -\frac{2}{3} \)

And for \( y \), we consistently found:
\[ y = 7 \]

### Conclusion

Since we have two different values for \( x \) and only one consistent value for \( y \), it is not possible to find a pair of values \( (x, y) \) that satisfies both matrices being equal.

Thus, the answer is that it is **not possible** to find values of \( x \) and \( y \) that make the matrices equal.

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