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In mathematics, when we refer to a system of equations, we're talking about a collection of two or more equations with a common set of variables. The goal is to find the point or points where all these equations hold true simultaneously. Think of it as searching for a crossroad in a city where all the different streets(intersecting lines) meet.

For example, the provided system involves two equations with two variables, x and y. In geometric terms, each equation represents a line on a graph, and we're looking for where these lines intersect, which would give us the solutions for x and y that satisfy all equations at once.

To solve these, we can use methods like substitution, elimination, or graphical interpretation, but here we focus on what's called Gauss-Jordan elimination. This method systematically transforms the system into a simpler form, ultimately showing us the point of intersection clearly or indicating that there might be other possibilities, like no intersection or an infinite number of intersections.

The Row Reduced Echelon Form (RREF) represents a matrix that has been transformed using Gauss-Jordan elimination to a state where we can easily read off the solutions to the system of equations it represents. Here are the hallmarks of RREF: each leading entry is 1, leading entries are to the right of those in rows above, rows with all zeros are at the bottom, and each leading entry is the only non-zero number in its column.

When we apply Gauss-Jordan elimination to the given system, our aim is to obtain such a form. By strategically adding, subtracting, and multiplying rows, we can clear out all other numbers in the columns containing the leading 1s. Through these operations, we reveal the essence of the system – whether it has one solution, no solution, or infinitely many solutions.

A system has dependent equations when one equation can be derived from another by multiplication or addition—essentially, they are different expressions of the same relationship. When we interpret the RREF of the matrix and see entire rows of zeros, this indicates dependency. The third equation in the system under consideration, after performing Gauss-Jordan elimination, vanishes and becomes a row of zeros.

This indicates that one of our equations was redundant—it didn't provide new information but rather repeated what was already known from the other equations. In terms of graphing these equations, dependent equations represent the same line or plane, so every point on the line (or plane) satisfies all equations. This is why dependent systems can lead to not just one intersection point, but an infinite number.

After transforming a system into RREF and finding dependent equations, we face the possibility that the system has infinite solutions. This happens when the equations describe the same geometric shape—like the same line or the same plane when dealing with more than two dimensions. In our scenario, because the RREF result had a row of zeros and the remaining rows were equivalent, we interpreted this as the system having an infinite number of solutions.

Imagine the original lines overlaying exactly one atop the other—every point where the first line exists is a point where the second line also exists, so there's not just one meeting place but countless. When the system is graphed, we don't see two lines intersecting at a point; instead, we see just one line. Every point on this line is a solution to the system, hence 'infinite solutions'. Understanding this concept is crucial, as it shows us that not all systems are neatly solvable with a single answer, and sometimes, they reveal an unbounded expanse of possibilities.