Question - Matrix Equation Solving

Given the equation:

\(\begin{pmatrix} 0 & 1 \\ y & 5 \end{pmatrix} + \begin{pmatrix} 4 & -1 \\ 6 & x \end{pmatrix} = \begin{pmatrix} 4 & 0 \\ x & 7 \end{pmatrix}\)

We can separate this into two equations by comparing corresponding elements:

1. \(0 + 4 = 4\)

2. \(1 - 1 = 0\)

3. \(y + 6 = x\)

4. \(5 + x = 7\)

From equation 1 and 2, we directly confirm that both hold true. Now, solving for \(x\) from equation 4:

From \(5 + x = 7\)

\(x = 7 - 5\)

Thus, \(x = 2\).

Now substitute \(x\) into equation 3:

\(y + 6 = 2\)

Thus, \(y = 2 - 6\)

So, \(y = -4\).

Final values: \(x = 2, y = -4\).