We are given a system of linear equations from a matrix equation. The problem is presented as follows:

$\begin{pmatrix} -2 & 1 \\ -3 & 4 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \begin{pmatrix} 15 \\ 5 \end{pmatrix} = \begin{pmatrix} 30 \\ 30 \end{pmatrix}$

Step 1: Simplify the equation

We start by isolating the matrix multiplication part:

$\begin{pmatrix} -2 & 1 \\ -3 & 4 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 30 \\ 30 \end{pmatrix} - \begin{pmatrix} 15 \\ 5 \end{pmatrix}$

This simplifies to:

$\begin{pmatrix} -2 & 1 \\ -3 & 4 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 15 \\ 25 \end{pmatrix}$

Step 2: Expand the matrix multiplication

Now, perform the matrix multiplication on the left-hand side: