Solve for **x** and **y** the following system of Linear Equations using Determinant of a matrix: a)

To solve the system of linear equations using the determinant of a matrix, we'll use Cramer's rule. Cramer's rule states that for a system of linear equations:

\[ \begin{cases} 
a_1x + b_1y = c_1 \\
a_2x + b_2y = c_2 
\end{cases} \]

We can represent the system in matrix form as \( AX = B \):

\[ A = \begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix} \]

The solution for \( X \) can be found using the determinants of matrices. The determinant of matrix \( A \) is given by:

\[ \text{det}(A) = a_1b_2 - a_2b_1 \]

Cramer's rule states:

\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} \]
\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} \]

where \( A_x \) and \( A_y \) are matrices obtained by replacing the columns of \( A \) with the column matrix \( B \).

\[ A_x = \begin{pmatrix} c_1 & b_1 \\ c_2 & b_2 \end{pmatrix} \]
\[ A_y = \begin{pmatrix} a_1 & c_1 \\ a_2 & c_2 \end{pmatrix} \]

Then, the determinants of \( A_x \) and \( A_y \) are:

\[ \text{det}(A_x) = c_1b_2 - c_2b_1 \]
\[ \text{det}(A_y) = a_1c_2 - a_2c_1 \]

Thus, the solutions for \( x \) and \( y \) are:

\[ x = \frac{c_1b_2 - c_2b_1}{a_1b_2 - a_2b_1} \]
\[ y = \frac{a_1c_2 - a_2c_1}{a_1b_2 - a_2b_1} \]

### Example
Let's apply this to a specific example:

\[ \begin{cases} 
2x + 3y = 5 \\
4x + 6y = 10 
\end{cases} \]

Here, \( a_1 = 2 \), \( b_1 = 3 \), \( c_1 = 5 \), \( a_2 = 4 \), \( b_2 = 6 \), \( c_2 = 10 \).

1. Calculate the determinant of \( A \):

\[ \text{det}(A) = 2 \cdot 6 - 4 \cdot 3 = 12 - 12 = 0 \]

Since the determinant of \( A \) is 0, the system of equations is either dependent or inconsistent. Cramer's rule cannot be used when the determinant is 0.

In this case, the equations are actually dependent because the second equation is a multiple of the first equation:

\[ 4x + 6y = 2(2x + 3y) = 2 \times 5 = 10 \]

Thus, any \( (x, y) \) that satisfies \( 2x + 3y = 5 \) will also satisfy \( 4x + 6y = 10 \). 

To find the general solution, solve for \( y \) in terms of \( x \):

\[ y = \frac{5 - 2x}{3} \]

Therefore, the solutions are:

\[ (x, y) = \left( x, \frac{5 - 2x}{3} \right) \]

If you have another system of linear equations you would like to solve using determinants, please provide them!

Would you like more details or have any questions?

Here are some related questions:

1. How do you determine if a system of equations has no solution, one solution, or infinitely many solutions using determinants?
2. What is Cramer's rule and how is it used in solving systems of linear equations?
3. Can you explain the conditions under which Cramer's rule is applicable?
4. How do you compute the determinant of a 3x3 matrix?
5. How can matrix methods be used to solve systems of linear equations?
6. What is the difference between consistent and inconsistent systems of equations?
7. How do you find the inverse of a matrix, and when is it used in solving systems of equations?
8. What are some real-world applications of solving systems of linear equations using matrices?

**Tip:** When solving systems of equations, always check the determinant first to see if a unique solution exists.

Learn how to solve a system of linear equations using the determinant of a matrix and Cramer's Rule. This problem involves understanding matrices, determinants, and applying Cramer's Rule to find solutions for x and y. Suitable for high school students studying algebra and matrix mathematics.

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