solve x-3y+3w=2, u=1, x-3y-z+7w+2u=6 —> using 1 by 5 matrix like matrix + matrix times y + matrix times w

To solve this system of equations using matrices, we can express the equations in matrix form. The given system is:

1. \( x - 3y + 3w = 2 \)
2. \( u = 1 \)
3. \( x - 3y - z + 7w + 2u = 6 \)

We want to represent this system using a matrix equation of the form:

\[
\mathbf{A} + \mathbf{B}y + \mathbf{C}w = \mathbf{D}
\]

### Step 1: Identify the unknowns and form matrices

The unknowns here are \( x, y, z, w, \) and \( u \). 

### Step 2: Express the equations in terms of these unknowns

We want to find suitable matrices such that:

\[
\begin{bmatrix} x \\ y \\ z \\ w \\ u \end{bmatrix} = \mathbf{A} + \mathbf{B}y + \mathbf{C}w
\]

### Step 3: Setting up the matrices

From the equations, observe how \( x, y, z, w, \) and \( u \) are related:

1. From \( x - 3y + 3w = 2 \), solve for \( x \):
   \[
   x = 3y - 3w + 2
   \]
   
2. From \( u = 1 \):
   \[
   u = 1
   \]
   
3. From \( x - 3y - z + 7w + 2u = 6 \), substitute \( u = 1 \):
   \[
   x - 3y - z + 7w + 2(1) = 6 \implies x - 3y - z + 7w = 4
   \]
   Using \( x = 3y - 3w + 2 \) from the first equation:
   \[
   (3y - 3w + 2) - 3y - z + 7w = 4
   \]
   Simplifying:
   \[
   -3w + 7w - z + 2 = 4 \implies 4w - z = 2 \implies z = 4w - 2
   \]

### Step 4: Express in matrix form

We can set up the matrices \( \mathbf{A}, \mathbf{B}, \) and \( \mathbf{C} \) to fit the desired matrix equation:

\[
\begin{bmatrix} x \\ y \\ z \\ w \\ u \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ -2 \\ 0 \\ 1 \end{bmatrix} + \begin{bmatrix} 3 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}y + \begin{bmatrix} -3 \\ 0 \\ 4 \\ 1 \\ 0 \end{bmatrix}w
\]

### Explanation:

1. **Constant Matrix \(\mathbf{A}\)**: \( \begin{bmatrix} 2 \\ 0 \\ -2 \\ 0 \\ 1 \end{bmatrix} \)
   - This reflects constants independent of \( y \) and \( w \):
     - \( x = 2 \)
     - \( y = 0 \)
     - \( z = -2 \)
     - \( w = 0 \)
     - \( u = 1 \)
   
2. **Matrix \(\mathbf{B}\) for \( y \)**: \( \begin{bmatrix} 3 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \)
   - Coefficients of \( y \) in the equations:
     - \( 3y \) affects \( x \)
     - \( y \) itself
     - \( z, w, u \) are unaffected by \( y \)
   
3. **Matrix \(\mathbf{C}\) for \( w \)**: \( \begin{bmatrix} -3 \\ 0 \\ 4 \\ 1 \\ 0 \end{bmatrix} \)
   - Coefficients of \( w \) in the equations:
     - \( -3w \) affects \( x \)
     - \( 4w \) affects \( z \)
     - \( w \) itself
     - \( y, u \) are unaffected by \( w \)

### Solution Summary:

Using the matrices, the solutions for \( x, y, z, w, \) and \( u \) can be expressed as:

\[
\begin{bmatrix} x \\ y \\ z \\ w \\ u \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ -2 \\ 0 \\ 1 \end{bmatrix} + \begin{bmatrix} 3 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}y + \begin{bmatrix} -3 \\ 0 \\ 4 \\ 1 \\ 0 \end{bmatrix}w
\]

### Questions for Further Exploration:

1. What happens if we introduce another variable \( t \) in the equations?
2. How would the solution change if \( u \) was a variable instead of being fixed at 1?
3. Can this approach be generalized for a system with more equations and unknowns?
4. How would we solve the system if non-linear terms were involved?
5. What are the implications of the rank of the coefficient matrix on the solution's existence and uniqueness?

### Tip:

Always double-check the consistency of your equations before forming matrices, ensuring that every equation is accounted for and correctly represented.

Learn how to solve a system of equations using matrices with step-by-step explanations. This problem involves matrix algebra and systems of equations, suitable for advanced high school students and beyond.

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