https://file.aiquickdraw.com/mathbot/file/17274125778247wsdp710.jpg

The problem in the image involves matrix multiplication and solving for variables. Here's a breakdown of the task:

Given:
- Matrix \( A = \begin{bmatrix} 2x & 3 \\ -2 & 2 \end{bmatrix} \)
- Matrix \( B = \begin{bmatrix} 4 & 5 \\ 2 & x \end{bmatrix} \)
- Matrix \( C = \begin{bmatrix} 2y & 26 \\ -2y & -6 \end{bmatrix} \)

We are asked to:
1. Compute \( A \cdot B = C \), and determine the values of \( x \) and \( y \).
2. Calculate \( 2x^2 - y \).

### Step 1: Matrix Multiplication \( A \cdot B = C \)

Multiplying matrices \( A \) and \( B \):

\[
A \cdot B = \begin{bmatrix} 2x & 3 \\ -2 & 2 \end{bmatrix} \cdot \begin{bmatrix} 4 & 5 \\ 2 & x \end{bmatrix} = \begin{bmatrix} (2x \cdot 4 + 3 \cdot 2) & (2x \cdot 5 + 3 \cdot x) \\ (-2 \cdot 4 + 2 \cdot 2) & (-2 \cdot 5 + 2 \cdot x) \end{bmatrix}
\]

This simplifies to:
\[
A \cdot B = \begin{bmatrix} 8x + 6 & 10x + 3x \\ -8 + 4 & -10 + 2x \end{bmatrix} = \begin{bmatrix} 8x + 6 & 13x \\ -4 & 2x - 10 \end{bmatrix}
\]

We now equate this result with matrix \( C \):
\[
\begin{bmatrix} 8x + 6 & 13x \\ -4 & 2x - 10 \end{bmatrix} = \begin{bmatrix} 2y & 26 \\ -2y & -6 \end{bmatrix}
\]

### Step 2: Set up equations from matrix equality
- From the first element: \( 8x + 6 = 2y \)
- From the second element: \( 13x = 26 \)
- From the third element: \( -4 = -2y \)
- From the fourth element: \( 2x - 10 = -6 \)

### Step 3: Solve for \( x \) and \( y \)

From \( 13x = 26 \):
\[
x = \frac{26}{13} = 2
\]

From \( -4 = -2y \):
\[
y = \frac{-4}{-2} = 2
\]

Substitute \( x = 2 \) into \( 2x - 10 = -6 \):
\[
2(2) - 10 = -6 \quad \text{(True, so this is correct)}
\]

Now substitute \( x = 2 \) into \( 8x + 6 = 2y \):
\[
8(2) + 6 = 2y \quad \Rightarrow \quad 16 + 6 = 2y \quad \Rightarrow \quad 22 = 2y \quad \Rightarrow \quad y = 11
\]

### Step 4: Compute \( 2x^2 - y \)

Substitute \( x = 2 \) and \( y = 11 \):
\[
2x^2 - y = 2(2)^2 - 11 = 2(4) - 11 = 8 - 11 = -3
\]

### Final Answer:
- \( x = 2 \)
- \( y = 11 \)
- \( 2x^2 - y = -3 \)

Would you like a detailed explanation of any part of the process?

Here are 5 questions to explore this topic further:
1. What are the steps involved in matrix multiplication?
2. How can you verify that two matrices are equal?
3. What are some common applications of matrix multiplication in real-world problems?
4. How can systems of equations be solved using matrix operations?
5. What are the properties of matrix operations (e.g., commutative, associative)?

**Tip:** When solving systems of equations from matrices, always check that the matrix dimensions are compatible for multiplication.

Determine the values of x and y such that the matrix product A.B = C, and calculate 2x^2 - y.

Learn how to solve for variables x and y by performing matrix multiplication and equating the resulting matrix with a given matrix C. This problem explores key concepts in matrix operations and algebra, suitable for students in Grades 10-12.

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