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A system of equations consists of two or more equations containing common variables. In the context of a 2x2 system, this generally involves two linear equations with two unknowns. These systems can arise in numerous real-life contexts, such as predicting financial outcomes or solving engineering problems.

Here’s how a system of equations can be set up:

• Identify the variables common to both equations - usually represented by letters like \(x\) and \(y\).
• Construct equations that relate these variables - often forming a straight line when graphed.

Understanding how to solve systems of equations is foundational because it sets the stage for more complex mathematical concepts like matrices and vector spaces.

A 2x2 matrix is a simple grid that consists of four numbers arranged into two rows and two columns. Each position in the matrix is significant and corresponds to a specific operation in mathematical expressions. For example, in a system of equations, a 2x2 matrix can represent the coefficients of the variables involved.

Consider the following matrix from the solution: \[A = \begin{pmatrix} 6 & -1 \ 3 & 2 \end{pmatrix}\]This matrix corresponds to the coefficients in the given system of equations.

The key properties of a 2x2 matrix include:

• Its determinant, which is calculated as \((ad-bc)\) for a matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \).

In this exercise, the determinant helps us find the inverse of the matrix, which is crucial for solving the system of equations using matrix inverses.

Matrix multiplication involves combining two matrices to produce a new matrix. This operation is essential when transitioning from the matrix equation \(AX = B\) to solve for \(X\), especially using matrix inverses. The process follows strict rules:

• To multiply a matrix \(A\) of size \(m \times n\) with another matrix \(B\) of size \(n \times p\), the number of columns in \(A\) must match the number of rows in \(B\).
• The resulting matrix will have dimensions \(m \times p\).
• Each element in the resulting matrix is calculated by taking the dot product of rows of the first matrix with columns of the second.

In the context of this exercise, we perform matrix multiplication between the inverse of a 2x2 matrix and another 2x1 matrix to find the values of \(x\) and \(y\). It's a powerful method not only because it provides a clear pathway to solutions, but also because it applies to broader mathematical problems beyond linear algebra.