https://file.aiquickdraw.com/mathbot/file/1726623183347887pauii.jpg

Let's solve the matrix equations shown in the image.

### Part (a)
The equation is:
\[
\begin{pmatrix} 
2x & -3 \\ 
1 & 9 
\end{pmatrix}
=
\begin{pmatrix} 
6 & -3 \\ 
1 & -y 
\end{pmatrix}
\]

For two matrices to be equal, their corresponding elements must be equal. Therefore, we have:
1. \( 2x = 6 \)
2. \( -3 = -3 \) (which is always true)
3. \( 1 = 1 \) (which is always true)
4. \( 9 = -y \)

Now, solve for \(x\) and \(y\):

1. \( 2x = 6 \Rightarrow x = \frac{6}{2} = 3 \)
2. \( 9 = -y \Rightarrow y = -9 \)

Thus, the values are \(x = 3\) and \(y = -9\).

---

### Part (b)
The equation is:
\[
\begin{pmatrix} 
x + 2 \\ 
x - y 
\end{pmatrix}
=
\begin{pmatrix} 
12 \\ 
5 
\end{pmatrix}
\]

Again, equating the corresponding elements gives us:
1. \( x + 2 = 12 \)
2. \( x - y = 5 \)

Now, solve these equations:
1. \( x + 2 = 12 \Rightarrow x = 12 - 2 = 10 \)
2. \( x - y = 5 \Rightarrow 10 - y = 5 \Rightarrow y = 10 - 5 = 5 \)

Thus, the values are \(x = 10\) and \(y = 5\).

---

### Final Answer:
- For part (a): \( x = 3 \), \( y = -9 \)
- For part (b): \( x = 10 \), \( y = 5 \)

Would you like further explanation on any of the steps? Here are some related questions to explore:

1. How do you determine if two matrices are equal?
2. What are the general rules for solving systems of equations using matrices?
3. How can you use matrix equations to solve real-world problems?
4. Can matrices of different sizes ever be equal? Why or why not?
5. What is the geometric interpretation of matrix operations?

**Tip:** When comparing matrices, always ensure that the corresponding elements in the same position are equal before solving for variables.

Determine the values of x and y from the following matrix equations: (a) [[2x, -3], [1, 9]] = [[6, -3], [1, -y]], and (b) [[x + 2], [x - y]] = [[12], [5]]

Solve matrix equations by equating corresponding elements to find the values of x and y. Explore algebraic manipulation and matrix equality in this detailed step-by-step guide. Suitable for Grades 9-10.

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