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A system of linear equations consists of multiple linear equations that are solved together in order to find values for a set of variables that satisfy all the equations simultaneously. Each equation represents a straight line in two-dimensional space or a plane in three-dimensional space, and the solution of a system is the point or set of points where these lines or planes intersect.

For example, in a system with three variables, the solution is the point in three-dimensional space where all three planes corresponding to the equations intercept. When utilizing Gaussian elimination to solve such a system, the goal is to use algebraic manipulation to simplify the equations to the point where the solution is clear. If the system is inconsistent, it means that there is no set of values for the variables that can satisfy all the equations, often because the planes represented by the equations do not intersect at a single point.

An augmented matrix is a compact way to represent a system of linear equations. It's constructed by writing the coefficients of the variables in a grid, with each row representing an equation and each column representing a variable. A vertical bar is used to separate the coefficients from the constants on the right-hand side of each equation.

In our exercise example, the augmented matrix makes it easy to perform row operations as part of the Gaussian elimination process. Operations like swapping rows, multiplying rows by non-zero constants, and adding multiples of one row to another are used to simplify the matrix, eventually leading to a form from which the solutions can easily be read.

Row-echelon form is a stage in Gaussian elimination where the augmented matrix has been simplified to a point where each successive row has more leading zeros than the previous row. The leading entry of each nonzero row (after the leading zeros) is strictly to the right of the leading entry in the row above. These leading entries are often made to be 1 (called a leading 1) to simplify the process further.

In this form, it becomes easy to 'read off' the solutions or to perform back-substitution to find the values of the variables. If a row has all zeros to the left of the bar and a nonzero number to the right, it indicates an inconsistency, meaning there is no solution. If there are at least as many equations as variables, and row-echelon form is reached without encountering such an inconsistency, one can typically find a unique solution to the system.