https://file.aiquickdraw.com/mathbot/file/1726710681964pgf1hkti.jpg

The image contains a matrix equation that needs to be solved. Here's the equation as it appears:

\[
2 \begin{pmatrix} 5 \\ -2 \end{pmatrix} - y \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}
\]

To solve for the variable \( y \), follow these steps:

1. **Distribute the scalar 2** to the first matrix:
   \[
   2 \begin{pmatrix} 5 \\ -2 \end{pmatrix} = \begin{pmatrix} 10 \\ -4 \end{pmatrix}
   \]
   So the equation now becomes:
   \[
   \begin{pmatrix} 10 \\ -4 \end{pmatrix} - y \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}
   \]

2. **Subtract the second matrix from both sides**:
   \[
   - y \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ -3 \end{pmatrix} - \begin{pmatrix} 10 \\ -4 \end{pmatrix}
   \]

3. **Calculate the subtraction**:
   \[
   \begin{pmatrix} 2 \\ -3 \end{pmatrix} - \begin{pmatrix} 10 \\ -4 \end{pmatrix} = \begin{pmatrix} 2 - 10 \\ -3 - (-4) \end{pmatrix} = \begin{pmatrix} -8 \\ 1 \end{pmatrix}
   \]
   So the equation becomes:
   \[
   - y \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} -8 \\ 1 \end{pmatrix}
   \]

4. **Solve for \( y \)** by comparing the coefficients:
   \[
   - y \cdot 3 = -8 \quad \text{and} \quad - y \cdot 1 = 1
   \]
   From the first equation:
   \[
   y = \frac{-8}{-3} = \frac{8}{3}
   \]
   From the second equation:
   \[
   y = \frac{-1}{1} = -1
   \]

   Since these two results for \( y \) are inconsistent, it suggests there may have been a mistake or this system has no solution under typical interpretation.

Let me know if you would like to go over this solution in more detail or if you have any questions!

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**Relative Questions:**

1. What does it mean if a system of matrix equations has no solution?
2. How do you interpret inconsistent results when solving for variables in matrix equations?
3. Could there be a different approach to solving this problem that may yield a solution?
4. What does the scalar multiplication in a matrix equation represent geometrically?
5. How can matrix equations be used in real-world applications?

**Tip:** Always double-check the consistency of equations when solving systems; inconsistent results often indicate either a mistake or a system with no solution.

Solve the matrix equation: 2 * (5, -2) - y * (3, 1) = (2, -3)

Learn how to solve the matrix equation 2(5, -2) - y(3, 1) = (2, -3) step-by-step. This problem involves scalar multiplication and solving for the variable y, suitable for high school students studying matrix algebra.

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