Here are the essential concepts you must grasp in order to answer the question correctly.
Gauss-Jordan Elimination
The Gauss-Jordan elimination method is a systematic procedure used to solve systems of linear equations. It involves transforming the augmented matrix of the system into reduced row echelon form (RREF) through a series of row operations. This method allows for easy identification of solutions, including unique solutions, no solutions, or infinitely many solutions, depending on the rank of the matrix.
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Infinitely Many Solutions
A system of equations has infinitely many solutions when there are fewer independent equations than variables, leading to free variables. In such cases, one or more variables can be expressed in terms of others, allowing for a general solution. For example, in a two-variable system, if y is free, the solution can be expressed as (x, y) = (expression in terms of y, y), where y can take any real number value.
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Parametric Representation
Parametric representation is a way to express the solutions of a system of equations using parameters to denote free variables. For instance, in a three-variable system with infinitely many solutions, if z is free, the solution set can be written as (x, y, z) = (expression in terms of z, expression in terms of z, z), where z can vary freely. This representation highlights the dependency of some variables on others and provides a clear understanding of the solution space.
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