Question - Matrix Addition and Equations

First, add the corresponding elements of the matrices:

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

Which simplifies to:

\(\begin{bmatrix} 4 & 0 \\ y + 6 & 5 + x \end{bmatrix}\)

Set the resulting matrix equal to the given matrix:

\(\begin{bmatrix} 4 & 0 \\ x & 7 \end{bmatrix}\)

This gives the equations:

1. \(y + 6 = x\)

2. \(5 + x = 7\)

From the second equation, solve for \(x\):

\(x = 7 - 5 = 2\)

Now substitute \(x = 2\) into the first equation:

\(y + 6 = 2 \implies y = 2 - 6 = -4\)

Thus, the solution is:

\(y = -4\), \(x = 2\)