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determina el valor de las variables que satisfacen la igualdad

The problem presented involves determining the values of variables that satisfy the equality for each system of linear equations provided. Let's break down and solve each part:

### Part a)
\[
-5 \begin{bmatrix} 2x - 1 \\ 3y \end{bmatrix} + 3 \begin{bmatrix} 4y - z \\ 2 \end{bmatrix} = \begin{bmatrix} -8 \\ 11 \\ -6 \end{bmatrix}
\]

1. Expand both matrices:
   \[
   -5 \begin{bmatrix} 2x - 1 \\ 3y \end{bmatrix} = \begin{bmatrix} -10x + 5 \\ -15y \end{bmatrix}
   \]
   \[
   3 \begin{bmatrix} 4y - z \\ 2 \end{bmatrix} = \begin{bmatrix} 12y - 3z \\ 6 \end{bmatrix}
   \]

2. Add the resulting matrices:
   \[
   \begin{bmatrix} -10x + 5 + 12y - 3z \\ -15y + 6 \end{bmatrix} = \begin{bmatrix} -8 \\ 11 \end{bmatrix}
   \]

3. Set up the equations:
   \[
   -10x + 12y - 3z = -8 - 5 \quad \text{(Resulting in the first equation)}
   \]
   \[
   -15y + 6 = 11 \quad \text{(Resulting in the second equation)}
   \]

This gives us a system of equations to solve for \(x\), \(y\), and \(z\).

### Part b)
\[
-2x \begin{bmatrix} 1 & 2 \\ -1 & -3 \end{bmatrix} = \begin{bmatrix} 4 & 2 \\ 4 & 2 \end{bmatrix}
\]

Expand and compare corresponding elements to find \(x\).

### Part c)
\[
\begin{bmatrix} x - 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 6 \\ z \end{bmatrix}
\]

Direct comparison will give the values of \(x\) and \(z\).

### Part d)
\[
\begin{bmatrix} 5 & -4 \\ -12 & 6 \end{bmatrix} \begin{bmatrix} y \\ z \end{bmatrix} = \begin{bmatrix} 7 \\ -28 \end{bmatrix}
\]

This is a system of linear equations that can be solved by matrix multiplication.

### Part e)
\[
\begin{bmatrix} 7 & 0 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} x & -2 \\ 4 & 3y \end{bmatrix}
\]

Compare and solve for \(x\) and \(y\).

Let me solve these systematically for you. Would you like me to go step by step for each equation?

Learn how to determine the values of variables that satisfy the given equalities in systems of linear equations. This problem involves matrix operations, specifically matrix multiplication and solving linear systems. Suitable for high school level students.

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