The problem requires us to find the matrix $M$ given the equation: $ \begin{bmatrix} 6 \\ -7 \end{bmatrix} + 3M = \begin{bmatrix} 12 \\ -19 \end{bmatrix} $

1. Problem Description

The problem requires us to find the matrix $M$ given the equation:

$\begin{bmatrix} 6 \\ -7 \end{bmatrix} + 3M = \begin{bmatrix} 12 \\ -19 \end{bmatrix}$

2. Solution Steps

First, we isolate $3M$ by subtracting the matrix $\begin{bmatrix} 6 \\ -7 \end{bmatrix}$ from both sides of the equation:

$3M = \begin{bmatrix} 12 \\ -19 \end{bmatrix} - \begin{bmatrix} 6 \\ -7 \end{bmatrix}$

$3M = \begin{bmatrix} 12 - 6 \\ -19 - (-7) \end{bmatrix}$

$3M = \begin{bmatrix} 6 \\ -12 \end{bmatrix}$

Next, we divide both sides of the equation by 3 to find $M$:

$M = \frac{1}{3} \begin{bmatrix} 6 \\ -12 \end{bmatrix}$

$M = \begin{bmatrix} \frac{6}{3} \\ \frac{-12}{3} \end{bmatrix}$

$M = \begin{bmatrix} 2 \\ -4 \end{bmatrix}$

3. Final Answer

$M = \begin{bmatrix} 2 \\ -4 \end{bmatrix}$