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Let’s go through the problems one by one from the image you uploaded.

### 1. Matrix Equation (Determine \(a\), \(b\), \(c\), and \(d\))
#### (a)
You are asked to solve the matrix equation and find the values of \(a + b\) and \(d + 2c\).
\[
\begin{bmatrix} 1 & 3 \\ -1 & a + b \end{bmatrix} = \begin{bmatrix} 4 & d + 2c \\ d & -2 \end{bmatrix}
\]
From the corresponding elements, we get the following equations:
- \(1 = 4\)
- \(3 = d + 2c\)
- \(-1 = d\)
- \(a + b = -2\)

From here, solve for \(d\), \(c\), and \(a + b\).

#### (b)
Next, you have to solve another matrix equation and determine \(a\), \(b\), \(c\), and \(d\):
\[
\begin{bmatrix} a - b & b + a \\ 3d + c & 2d - c \end{bmatrix} = \begin{bmatrix} 8 & 1 \\ 7 & 6 \end{bmatrix}
\]
This gives you four more equations:
- \(a - b = 8\)
- \(b + a = 1\)
- \(3d + c = 7\)
- \(2d - c = 6\)

Solve these four equations simultaneously to find \(a\), \(b\), \(c\), and \(d\).

### 2. Solving Systems of Linear Equations
#### (a)
Solve the following system using substitution or elimination:
\[
\begin{aligned}
x - 2y &= 5 \\
-2x + 4y &= 7
\end{aligned}
\]

#### (b)
Another system of linear equations:
\[
\begin{aligned}
-x + 2y &= -5 \\
3x - 6y &= 15
\end{aligned}
\]

### 3. Solve the Systems of Equations from the Given Matrices
You are given augmented matrices in row echelon form (REF). You need to solve for the unknowns:
#### (a)
\[
\begin{bmatrix} 1 & 0 & 0 & 3 \\ 0 & 2 & 0 & 4 \\ 0 & 0 & 1 & 5 \end{bmatrix}
\]
#### (b)
\[
\begin{bmatrix} 1 & 3 & 0 & 0 & 4 \\ 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}
\]
#### (c)
\[
\begin{bmatrix} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 6 \\ 0 & 0 & 1 & 2 \end{bmatrix}
\]
#### (d)
\[
\begin{bmatrix} 1 & 6 & 0 & 0 & 2 & -3 \\ 0 & 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}
\]

### 4. Solve the Linear System in Reduced Row Echelon Form (RREF)
You are asked to solve the following system:
\[
\begin{aligned}
2x_1 + 4x_2 - 10x_3 + 6x_4 + 12x_5 &= 28 \\
2x_1 + 4x_2 - 5x_3 + 6x_4 - 5x_5 &= -1 \\
-x_3 + 7x_5 &= 12
\end{aligned}
\]

Transform this system into an augmented matrix and then apply Gaussian elimination to reduce it to RREF.

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Would you like to start with a specific question from this set, or should I provide the detailed solutions step by step for all questions? 

Here are some related questions that could help extend this content:
1. How do you find the inverse of a matrix, and when is it applicable?
2. What are the main differences between row echelon form and reduced row echelon form?
3. How do you check if a system of linear equations is consistent or inconsistent?
4. What methods can be used to solve a system of linear equations other than Gaussian elimination?
5. How can we interpret solutions in terms of free variables in a homogeneous system?

Tip: When solving systems of equations, checking for consistency early (through determinants or row reduction) can save you time in identifying if the system has no solutions or infinite solutions.

Solve matrix equations, systems of linear equations, and reduce augmented matrices to row echelon form.

Learn how to solve matrix equations, systems of linear equations, and reduce augmented matrices to row echelon form. This step-by-step guide covers matrix operations, Gaussian elimination, and solving systems of linear equations. Ideal for students in Grades 10-12 or those taking college-level linear algebra.

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