To solve the matrix equation \(\begin{pmatrix}5 & -6 \\ 2 & -7\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}7 \\ -11\end{pmatrix}\), we can perform matrix multiplication:

\(\begin{pmatrix}5 & -6 \\ 2 & -7\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}5x - 6y \\ 2x - 7y\end{pmatrix}\)

Now we have the system of equations:

\(5x - 6y = 7\)

\(2x - 7y = -11\)

We can solve this system of equations to find the value of \(y\). Let's use the method of substitution:

From the first equation, we can isolate \(x\):

\(5x = 6y + 7\)

\(x = \frac{6y + 7}{5}\)

Now substitute this expression for \(x\) into the second equation:

\(2\left(\frac{6y + 7}{5}\right) - 7y = -11\)

Simplify the equation:

\(\frac{12y + 14}{5} - 7y = -11\)

Multiply both sides by 5 to eliminate the fraction:

12y + 14 - 35y = -55

Combine like terms:

-23y + 14 = -55

Subtract 14 from both sides:

-23y = -69

Divide by -23:

y = 3