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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:

\[
\begin{aligned}
-2x_1 + x_2 &= 15 \\
-3x_1 + 4x_2 &= 25
\end{aligned}
\]

### Step 3: Solve the system of equations
We can solve this

Find x₁ and x₂ in the matrix equation: [[-2, 1], [-3, 4]] * [x₁, x₂] + [15, 5] = [30, 30].

Learn how to solve a system of linear equations using matrix operations. This problem involves multiplying matrices, simplifying the system, and solving for x₁ and x₂. Suitable for students in Grades 9-12.

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