Solve the following system by the inverse matrix method, if possible. If the inverse matrix method doesn't apply, use any other method to determine if the system is inconsistent or dependent. If there is a solution, write your answer in the format (x,y)

{2x+4y=−6        −6x−12y=18

To solve the system of equations using the inverse matrix method, we first need to write the system in matrix form:

\[
\begin{cases}
2x + 4y = -6 \\
-6x - 12y = 18
\end{cases}
\]

This can be represented as:

\[
\begin{bmatrix}
2 & 4 \\
-6 & -12
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}
=
\begin{bmatrix}
-6 \\
18
\end{bmatrix}
\]

Let's denote the coefficient matrix as \(A\) and the constants vector as \(B\):

\[
A = \begin{bmatrix}
2 & 4 \\
-6 & -12
\end{bmatrix}, \quad B = \begin{bmatrix}
-6 \\
18
\end{bmatrix}
\]

Next, we need to check if \(A\) is invertible. A matrix is invertible if its determinant is non-zero.

The determinant of \(A\) is calculated as follows:

\[
\text{det}(A) = (2)(-12) - (4)(-6) = -24 + 24 = 0
\]

Since the determinant of \(A\) is 0, \(A\) is not invertible. Therefore, we cannot use the inverse matrix method to solve this system.

Since the inverse matrix method is not applicable, we will use another method to determine if the system is inconsistent or dependent. We can use row reduction to echelon form (Gaussian elimination).

Starting with the augmented matrix:

\[
\left[\begin{array}{cc|c}
2 & 4 & -6 \\
-6 & -12 & 18
\end{array}\right]
\]

Perform row operations to simplify:

1. Divide the first row by 2:

\[
\left[\begin{array}{cc|c}
1 & 2 & -3 \\
-6 & -12 & 18
\end{array}\right]
\]

2. Add 6 times the first row to the second row:

\[
\left[\begin{array}{cc|c}
1 & 2 & -3 \\
0 & 0 & 0
\end{array}\right]
\]

The second row of the augmented matrix now consists of all zeros, which indicates that the second equation is a multiple of the first equation. Therefore, the system is dependent, meaning there are infinitely many solutions.

To express the solution set, we can parameterize \(y\) as \(t\) (where \(t\) is any real number), and solve for \(x\):

\[
x + 2y = -3 \implies x + 2t = -3 \implies x = -3 - 2t
\]

Thus, the solution set can be written as:

\[
(x, y) = (-3 - 2t, t) \quad \text{for any real number } t
\]

This means that the system has infinitely many solutions, which can be represented in the form \((x, y)\) as \((-3 - 2t, t)\).

Would you like further details or have any questions?

Here are some related questions to explore:

1. What is the significance of a matrix having a zero determinant?
2. How do you determine if a system of equations is consistent or inconsistent?
3. What are the different methods to solve a system of linear equations?
4. How can you check if two linear equations are dependent?
5. What are the conditions for a unique solution in a system of linear equations?
6. How do you interpret a system of equations geometrically?
7. What is Gaussian elimination, and how does it work?
8. How can you find the general solution for a dependent system of equations?

**Tip:** When solving systems of equations, always check the determinant of the coefficient matrix to quickly determine if the inverse matrix method is applicable.

Learn how to solve the system of linear equations 2x + 4y = -6 and -6x - 12y = 18 using matrix operations and determinants. This problem explores the concept of matrix invertibility and Gaussian elimination method. Suitable for students in Grades 11-12.

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