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The given system of equations is:

\[
\begin{aligned}
1. & \quad 2x + 5y = 14 \\
2. & \quad -8x - 20y = -56
\end{aligned}
\]

We can solve this system using the inverse matrix method. To do this, we first express the system in matrix form \( AX = B \), where:

\[
A = \begin{pmatrix} 2 & 5 \\ -8 & -20 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 14 \\ -56 \end{pmatrix}
\]

We need to check if matrix \( A \) has an inverse. The inverse of a \( 2 \times 2 \) matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) exists if the determinant \( \det(A) \) is non-zero, where:

\[
\det(A) = ad - bc
\]

For matrix \( A = \begin{pmatrix} 2 & 5 \\ -8 & -20 \end{pmatrix} \), the determinant is:

\[
\det(A) = (2)(-20) - (5)(-8) = -40 + 40 = 0
\]

Since the determinant is \( 0 \), the matrix \( A \) is singular, meaning it does not have an inverse.

Because the inverse matrix method cannot be applied, we must determine if the system is inconsistent or dependent by analyzing the equations. Notice that equation 2 is a multiple of equation 1:

\[
-8x - 20y = -4(2x + 5y)
\]

This means the second equation is just a scalar multiple of the first, implying that the system is dependent. Therefore, the system has infinitely many solutions, as both equations represent the same line.

Would you like more details or have any further questions?

Here are some related questions:
1. How do you find the inverse of a matrix when it exists?
2. What is the geometric interpretation of dependent systems of equations?
3. How can you solve a dependent system using substitution?
4. What does it mean when the determinant of a matrix is zero?
5. Can matrix methods solve inconsistent systems?

**Tip:** If the determinant of a matrix is zero, the system is either dependent (infinite solutions) or inconsistent (no solutions).

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.

This problem demonstrates how to solve a system of linear equations using the inverse matrix method. It explains key concepts of matrices, determinants, and how to identify dependent or inconsistent systems. Suitable for high school students (Grades 10-12).

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