https://file.aiquickdraw.com/mathbot/file/1725552087272u1898r85.jpg

Let's start by analyzing and solving the problems in the image:

### Problem 1:
**Solve the system of linear equations (SPLDV) using the matrix inverse method.**

The provided systems are:

- \( a) \begin{cases} 2x + y = 5 \\ x + 2y = 6 \end{cases} \)
- \( b) \begin{cases} 2x + y = 3 \\ x + 2y = 4 \end{cases} \)
- \( c) \begin{cases} 3x + 2y = 9 \\ 2x + 3y = 8 \end{cases} \)
- \( d) \begin{cases} 2x + 3y = 4 \\ 3x + 2y = 5 \end{cases} \)

We will solve these systems using the matrix inverse method. In general, the matrix form of a system of linear equations \( A \vec{x} = \vec{b} \) can be written as:
\[
\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix} =
\begin{pmatrix} b_1 \\ b_2 \end{pmatrix}
\]

To solve for \( \vec{x} \), we compute the inverse of the coefficient matrix \( A \) and multiply it by \( \vec{b} \):
\[
\vec{x} = A^{-1} \vec{b}
\]

Let's calculate this for each system.

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### Problem 2:
**For each SPLDV provided, analyze if the system has many solutions, no solutions, or a unique solution, and plot the system on the Cartesian plane.**

- \( a) \begin{cases} 3x + y = 7 \\ 3x - 2y = 10 \end{cases} \)
- \( b) \begin{cases} 6x + 2y = 10 \\ 3x + y = 5 \end{cases} \)
- \( c) \begin{cases} 3x + y = 6 \\ 3x + y = 6 \end{cases} \)

You need to determine whether the system is consistent (has a solution), dependent (infinite solutions), or inconsistent (no solution). This can be done by analyzing the determinant of the matrix or by plotting

Solve systems of linear equations (SPLDV) using the matrix inverse method and analyze their solutions.

Learn how to solve systems of linear equations (SPLDV) using the matrix inverse method. This problem involves solving multiple systems using matrix inversion, determinant analysis, and plotting solutions on a Cartesian plane. Ideal for students in Grades 10-12.

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