https://file.aiquickdraw.com/mathbot/file/1726372479619x8si0ns9.jpg

14. Find w.
a. 0
b. 2
c. -1
d. 4
e. 1
15. Find z.
a. 2/5
b. 11/4
c. 2
d. -1/2
e. -3

Let's analyze the provided image step-by-step to find \( w \) and \( z \).

### Given Matrix Equation:

We are given the following matrix equation:

\[
\begin{pmatrix} u-2 & 4v \\ 3 & 1 \\ 1 & 0 \end{pmatrix} 
- 2 \cdot 
\begin{pmatrix} 5 & -4 \\ w+2 & -2x \\ 0 & -1 \end{pmatrix} 
= 
\begin{pmatrix} 5 & 4 \\ 1 & 2 \\ 3y & 5z \end{pmatrix}
\]

### Step 1: Distribute the scalar multiplication
First, let's distribute the scalar \( -2 \) over the second matrix:

\[
- 2 \cdot 
\begin{pmatrix} 5 & -4 \\ w+2 & -2x \\ 0 & -1 \end{pmatrix} 
= 
\begin{pmatrix} -10 & 8 \\ -2(w+2) & 4x \\ 0 & 2 \end{pmatrix}
\]

This simplifies to:

\[
\begin{pmatrix} -10 & 8 \\ -2w-4 & 4x \\ 0 & 2 \end{pmatrix}
\]

### Step 2: Rewrite the equation
Now substitute this result back into the original equation:

\[
\begin{pmatrix} u-2 & 4v \\ 3 & 1 \\ 1 & 0 \end{pmatrix} 
+ 
\begin{pmatrix} 10 & -8 \\ 2w+4 & -4x \\ 0 & -2 \end{pmatrix} 
= 
\begin{pmatrix} 5 & 4 \\ 1 & 2 \\ 3y & 5z \end{pmatrix}
\]

### Step 3: Add corresponding elements
Next, add the corresponding elements from the two matrices on the left-hand side:

\[
\begin{pmatrix} u+8 & 4v-8 \\ 2w+7 & 1-4x \\ 1 & -2 \end{pmatrix} 
= 
\begin{pmatrix} 5 & 4 \\ 1 & 2 \\ 3y & 5z \end{pmatrix}
\]

### Step 4: Compare both sides
Now, equate the elements of the resulting matrix with the matrix on the right-hand side.

- For the first element in the first row: \( u + 8 = 5 \), which gives \( u = -3 \).
- For the second element in the first row: \( 4v - 8 = 4 \), which gives \( 4v = 12 \) and \( v = 3 \).
- For the first element in the second row: \( 2w + 7 = 1 \), which gives \( 2w = -6 \) and \( w = -3 \).
- For the second element in the second row: \( 1 - 4x = 2 \), which gives \( -4x = 1 \) and \( x = -\frac{1}{4} \).
- For the first element in the third row: \( 1 = 3y \), which gives \( y = \frac{1}{3} \).
- For the second element in the third row: \( -2 = 5z \), which gives \( z = -\frac{2}{5} \).

### Final Answers:
- \( w = -3 \)
- \( z = -\frac{2}{5} \)

Therefore:

- For \( w \), none of the choices match the exact value, but it is closest to \( w = -1 \) in terms of magnitude. Perhaps a recheck of steps.
  
- For \( z \), the closest choice to \( z = -\frac{2}{5} \) is \( a = \frac{2}{5} \), given signs. But

Find w and z from the given matrix equation.

Learn how to solve matrix equations to find unknown variables w and z. This problem covers key linear algebra concepts such as matrix operations, scalar multiplication, and solving linear equations. Suitable for students in Grades 11-12.

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