5x - 4x + 3x +2x +4
-x + x + 3x + 3x = 1
4x -3x+6x+5x = 5
-2x +2x +6x +6x = 2

solve the system in the form of three 4x1 matrices

To solve the system of equations and express the solution in the form of three 4x1 matrices, let's break it down step by step.

### 1. Simplify each equation

#### Equation 1:
\[ 5x - 4x + 3x + 2x + 4 = 0 \]
Simplify:
\[ (5x - 4x + 3x + 2x) + 4 = 0 \]
\[ 6x + 4 = 0 \]
\[ 6x = -4 \]
\[ x = -\frac{2}{3} \]

#### Equation 2:
\[ -x + x + 3x + 3x = 1 \]
Simplify:
\[ (-x + x + 3x + 3x) = 1 \]
\[ 6x = 1 \]
\[ x = \frac{1}{6} \]

#### Equation 3:
\[ 4x - 3x + 6x + 5x = 5 \]
Simplify:
\[ (4x - 3x + 6x + 5x) = 5 \]
\[ 12x = 5 \]
\[ x = \frac{5}{12} \]

#### Equation 4:
\[ -2x + 2x + 6x + 6x = 2 \]
Simplify:
\[ (-2x + 2x + 6x + 6x) = 2 \]
\[ 12x = 2 \]
\[ x = \frac{1}{6} \]

### 2. Express the solution in the form of three \(4 \times 1\) matrices

We can organize the simplified system of equations in matrix form. The first matrix represents the coefficients, the second matrix represents the variables, and the third matrix represents the constants.

The general form of a system of linear equations is:

\[
Ax = B
\]

Where:
- \(A\) is the coefficient matrix.
- \(x\) is the variable matrix.
- \(B\) is the constant matrix.

For the given system, we can write:

\[
A =
\begin{pmatrix}
6 \\
6 \\
12 \\
12
\end{pmatrix},
\quad
x =
\begin{pmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{pmatrix},
\quad
B =
\begin{pmatrix}
-4 \\
1 \\
5 \\
2
\end{pmatrix}
\]

Since each equation only involves \(x\), we combine all equations' solutions for each \(x\):

- \(x_1 = -\frac{2}{3}\)
- \(x_2 = \frac{1}{6}\)
- \(x_3 = \frac{5}{12}\)
- \(x_4 = \frac{1}{6}\)

Thus, the solution for \(x\) is:

\[
x =
\begin{pmatrix}
-\frac{2}{3} \\
\frac{1}{6} \\
\frac{5}{12} \\
\frac{1}{6}
\end{pmatrix}
\]

### 3. Summary of the solution in matrix form:
- Coefficient matrix \(A\):
\[
A =
\begin{pmatrix}
6 \\
6 \\
12 \\
12
\end{pmatrix}
\]

- Variable matrix \(x\):
\[
x =
\begin{pmatrix}
x_1 \\
x_2 \\
x_3 \\
x_4
\end{pmatrix}
\]

- Constant matrix \(B\):
\[
B =
\begin{pmatrix}
-4 \\
1 \\
5 \\
2
\end{pmatrix}
\]

Do you want further details or have any questions?

### 5 Related Questions:
1. How are matrices used to solve systems of linear equations?
2. What is the determinant of a matrix, and how does it relate to the solvability of a system?
3. Can systems of equations be solved using inverse matrices?
4. How do row operations affect the solution of a system of equations?
5. How would the solution change if additional variables were introduced?

### Tip: When solving systems of linear equations, checking for linear dependence between the equations can help determine if a unique solution exists.

The query submitted by the user: 5x - 4x + 3x + 2x + 4 = 0; -x + x + 3x + 3x = 1; 4x - 3x + 6x + 5x = 5; -2x + 2x + 6x + 6x = 2. Solve the system in the form of three 4x1 matrices.

This problem involves solving a system of linear equations and representing it as three 4x1 matrices. It covers algebraic simplification and matrix representation, suitable for students in Grades 10-12. Key concepts include matrix representation of linear systems and solving equations using matrix methods.

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