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A system of linear equations consists of two or more linear equations involving the same set of variables. In our original problem, we have two equations with two variables, \(x\) and \(y\):\

• \(-2x + 3y = 5\)
• \(6x + 7y = 4\)

When solving a system of equations, we aim to find values of \(x\) and \(y\) that satisfy all the equations involved. There are various methods for solving these systems, including substitution, elimination, and using matrices, which is the focus here.\
By converting these equations into matrix form, we can leverage the power of linear algebra techniques to simplify the problem-solving process.

Matrix dimensions refer to the number of rows and columns a matrix has. The matrix derived from our system of equations is usually an important visual and computational tool. In this case, the matrix is created from the coefficients of the system’s variables \((-2, 3)\) and \((6, 7)\):
\[\begin{pmatrix}-2 & 3 \6 & 7 \\end{pmatrix}\
\] To determine the dimensions of a matrix, we count the rows and columns it contains. For this matrix:

• There are 2 rows.
• There are 2 columns.

Thus, the matrix dimensions are \(2 \times 2\). Understanding the correct dimensions is crucial when performing matrix operations like addition, multiplication, or checking compatibility with other matrices.

Linear algebra is a branch of mathematics that focuses on vectors, matrices, and operations involving them. It provides tools for working with systems of linear equations and helps in understanding how solutions can be represented and manipulated.
In our exercise, linear algebra helps us use matrices to represent systems of equations more compactly. The concept of an augmented matrix is an extension of this tool, where we include an additional column representing the constants from the linear equations. This allows us to visualize and perform operations on equations as a whole.

• Matrices simplify complex systems into manageable forms.
• This helps apply systematic methods to solve them, such as reduction to row-echelon form or Gaussian elimination.

Understanding these processes requires grasping matrix operations and how they correspond to operations on equations. Linear algebra opens up powerful computational possibilities, making equation solving both efficient and scalable.