https://file.aiquickdraw.com/mathbot/file/1727008116090gispmnfj.jpg

The image contains a system of linear equations in matrix form that can be described as follows:

The matrix equation is given by:

\[
\begin{pmatrix}
4 & 3x - y \\
18 & 6
\end{pmatrix}
= 
\begin{pmatrix}
4 & 12 \\
xy & 6
\end{pmatrix}
\]

We are tasked with determining the values of \(x\) and \(y\).

### Step-by-step Solution:

1. **Compare the elements of the matrices:**
   Since the two matrices are equal, corresponding elements must be equal.

   - First row, first column: \( 4 = 4 \) (This is correct).
   - First row, second column: \( 3x - y = 12 \).
   - Second row, first column: \( 18 = xy \).
   - Second row, second column: \( 6 = 6 \) (This is also correct).

2. **Solve the system of equations:**
   Now, solve for \(x\) and \(y\) using the two equations:
   
   - \( 3x - y = 12 \)  (Equation 1)
   - \( xy = 18 \) (Equation 2)

### Solving the System:

1. From Equation 1: \( y = 3x - 12 \).
   
2. Substitute \( y = 3x - 12 \) into Equation 2:

   \[
   x(3x - 12) = 18
   \]
   
   Expand the equation:

   \[
   3x^2 - 12x = 18
   \]

   Rearrange into a standard quadratic equation:

   \[
   3x^2 - 12x - 18 = 0
   \]

   Divide through by 3:

   \[
   x^2 - 4x - 6 = 0
   \]

3. Solve the quadratic equation using the quadratic formula:

   \[
   x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-6)}}{2(1)}
   \]

   Simplifying:

   \[
   x = \frac{4 \pm \sqrt{16 + 24}}{2} = \frac{4 \pm \sqrt{40}}{2} = \frac{4 \pm 2\sqrt{10}}{2}
   \]

   \[
   x = 2 \pm \sqrt{10}
   \]

   Thus, \(x\) can be \( x = 2 + \sqrt{10} \) or \( x = 2 - \sqrt{10} \).

4. Substitute these values of \(x\) back into \( y = 3x - 12 \) to find \(y\):

   - For \( x = 2 + \sqrt{10} \):
     \[
     y = 3(2 + \sqrt{10}) - 12 = 6 + 3\sqrt{10} - 12 = 3\sqrt{10} - 6
     \]

   - For \( x = 2 - \sqrt{10} \):
     \[
     y = 3(2 - \sqrt{10}) - 12 = 6 - 3\sqrt{10} - 12 = -6 - 3\sqrt{10}
     \]

### Final Answer:
The solutions are:

- \( x = 2 + \sqrt{10} \), \( y = 3\sqrt{10} - 6 \)
- \( x = 2 - \sqrt{10} \), \( y = -6 - 3\sqrt{10} \)

Would you like more details on any part of this process?

### Related Questions:
1. How do you solve a system of linear equations using substitution?
2. What are the properties of matrix equality?
3. How can the quadratic formula be derived?
4. What does solving a quadratic equation geometrically look like?
5. What methods can be used to solve non-linear systems?

### Tip:
Always check the solutions by substituting back into the original equations to ensure they satisfy both.

Determine the values of x and y from the following matrix equation: [4, 3x - y; 18, 6] = [4, 12; xy, 6].

Learn how to solve a matrix equation involving variables x and y. This problem covers linear equations, matrix equality, and quadratic equations. Suitable for students in Grades 10-12.

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