Determine values of the variables that will make the following equation true, if possible. If not, state "Not Possible".  2((2W,-2X),(-y,4z)) +5 ((w,x), (4y,4z)) = ((w-48,2), (108,196))

We are given the matrix equation:

\[
2\begin{pmatrix} 2W & -2X \\ -y & 4z \end{pmatrix} + 5 \begin{pmatrix} w & x \\ 4y & 4z \end{pmatrix} = \begin{pmatrix} w - 48 & 2 \\ 108 & 196 \end{pmatrix}
\]

We need to find values of \( w \), \( x \), \( y \), and \( z \) that satisfy this equation.

### Step 1: Expand the left-hand side

First, let's multiply the scalar constants into the matrices on the left-hand side:

\[
2\begin{pmatrix} 2W & -2X \\ -y & 4z \end{pmatrix} = \begin{pmatrix} 4W & -4X \\ -2y & 8z \end{pmatrix}
\]
\[
5 \begin{pmatrix} w & x \\ 4y & 4z \end{pmatrix} = \begin{pmatrix} 5w & 5x \\ 20y & 20z \end{pmatrix}
\]

Now add these two matrices together:

\[
\begin{pmatrix} 4W & -4X \\ -2y & 8z \end{pmatrix} + \begin{pmatrix} 5w & 5x \\ 20y & 20z \end{pmatrix} = \begin{pmatrix} 4W + 5w & -4X + 5x \\ -2y + 20y & 8z + 20z \end{pmatrix}
\]
\[
= \begin{pmatrix} 4W + 5w & -4X + 5x \\ 18y & 28z \end{pmatrix}
\]

### Step 2: Set up the system of equations

Now, equate this result to the matrix on the right-hand side of the original equation:

\[
\begin{pmatrix} 4W + 5w & -4X + 5x \\ 18y & 28z \end{pmatrix} = \begin{pmatrix} w - 48 & 2 \\ 108 & 196 \end{pmatrix}
\]

This gives us four equations by comparing corresponding elements:

1. \( 4W + 5w = w - 48 \)
2. \( -4X + 5x = 2 \)
3. \( 18y = 108 \)
4. \( 28z = 196 \)

### Step 3: Solve the system of equations

#### Equation 3: \( 18y = 108 \)
\[
y = \frac{108}{18} = 6
\]

#### Equation 4: \( 28z = 196 \)
\[
z = \frac{196}{28} = 7
\]

#### Equation 1: \( 4W + 5w = w - 48 \)
Simplify:
\[
4W = w - 48 - 5w
\]
\[
4W = -4w - 48
\]
\[
W = -w - 12
\]

#### Equation 2: \( -4X + 5x = 2 \)
\[
-4X = 2 - 5x
\]
\[
X = \frac{5x - 2}{4}
\]

### Final Answer:
- \( W = -w - 12 \)
- \( X = \frac{5x - 2}{4} \)
- \( y = 6 \)
- \( z = 7 \)

These are the values and relationships between the variables that will make the equation true.

Would you like any details on how these steps were carried out, or do you have any questions?

Here are 5 related questions you might explore:
1. How do you solve systems of linear equations?
2. What are the properties of matrix addition and scalar multiplication?
3. How does matrix multiplication differ from element-wise multiplication?
4. How do determinants and inverses apply to matrix equations?
5. Can matrix equations always be solved?

**Tip:** Always simplify each matrix step-by-step to avoid errors, especially when dealing with variables.

Determine values of the variables that will make the following equation true, if possible. If not, state 'Not Possible'.  2((2W,-2X),(-y,4z)) + 5 ((w,x), (4y,4z)) = ((w-48,2), (108,196))

Explore how to solve the matrix equation 2((2W,-2X),(-y,4z)) + 5((w,x), (4y,4z)) = ((w-48,2),(108,196)) using algebraic techniques. The solution involves matrix addition, scalar multiplication, and solving a system of linear equations for variables w, x, y, and z. Suitable for Grades 10-12.

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