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A system of linear equations is a collection of one or more linear equations involving the same variables. For example, equations like \(2x + 3y = 5\) and \(4x - y = 2\) form a system because they both involve the variables \(x\) and \(y\). Each equation in the system represents a line, and the solution to the system is the point or points where the lines intersect.

When dealing with these systems, we aim to find all values of the variables that satisfy all the equations in the system simultaneously. Several methods can be used to solve these systems, including graphing, substitution, elimination, and matrix operations. The latter, often considered the most systematic and general method, involves writing the system in matrix form and using row operations to simplify the matrix, a process known as Gaussian elimination. The goal is to obtain a row-reduced echelon form of the matrix from which the solutions are straightforward to read.

An augmented matrix is an array of numbers used to represent a system of linear equations. It combines the coefficients of the variables and the constants from the equations into one matrix. For instance, the equations \(x + 2y = 4\) and \(3x - y = 2\) can be represented by the augmented matrix \[\left[\begin{array}{cc|c}1 & 2 & 4 \3 & -1 & 2\end{array}\right]\].

The matrix is divided into two sections by a vertical line: on the left are the coefficients of the variables, and on the right are the constants from the right-hand side of the equations. The advantage of using an augmented matrix is that it simplifies the manipulation of a system of equations, especially when using methods like Gaussian elimination to find solutions.

Constructing an Augmented Matrix

To construct an augmented matrix from a system of linear equations, list the coefficients of the variables in the order they appear in the equations, followed by the constants on the right side of the vertical bar. The rows of the matrix correspond to the individual equations of the system.

A system of linear equations has a unique solution when there is exactly one set of values for the variables that satisfies all the equations. This occurs when the lines represented by the equations all intersect at a single point, meaning for two variables, the lines cross at one point, and for three variables, the planes intersect in a single line.

In terms of matrices, a unique solution is present when the augmented matrix of a system, after being converted to row-reduced echelon form, has leading ones (pivot positions) in every row and no row is made up entirely of zeros. This means each variable can be solved with a particular value, with no need for parameters or arbitrary choices. When solving a system of equations, finding a unique solution is an ideal scenario as it provides a clear and definite answer to the problem presented.

Signs of a Unique Solution

Indicators of a unique solution, when dealing with augmented matrices, include the presence of a diagonal of leading ones and a lack of free variables or contradictory equations, such as \(0=1\).