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.