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Understanding how to solve systems of linear equations is crucial for dealing with problems that have multiple variables interacting with each other. A system of linear equations consists of two or more linear equations with the same variables. For example, in the given exercise, we're looking at a system with two equations, each involving the variables x and y. The goal when solving such a system is to find the values of these variables that satisfy all equations simultaneously.

The addition (or elimination) method is one of the techniques used to solve such systems. It involves adding or subtracting the equations from one another to eliminate one variable, making it possible to solve for the other variable. However, it's essential to recognize situations where this method gives us special results, such as when the equations are identical or when they are parallel, which leads to either infinite solutions or no solution at all.

Simplifying equations is a fundamental skill in algebra that helps to reveal the structure of the system you're working with. It involves reducing equations to their simplest form by performing valid algebraic operations, such as addition, subtraction, multiplication, division, and factoring.

As seen in the exercise, the second equation was simplified by dividing every term by 3. This process showed that the original system was, in fact, not two distinct equations but rather two identical equations written in a slightly different form. Simplification made it apparent that we're dealing with a special kind of system with an interesting set of solutions. As students, identifying such cases early on can save time and guide towards the correct method of finding the solution.

The concept of infinite solutions arises in situations where our system of equations does not narrow down to a single unique solution. Instead, any value chosen for the variables will satisfy all equations in the system.

In the case of the provided exercise, after simplification and attempting the addition method, we end up with the equation 0=0. This result is correct for all values of x and y and indicates that the two equations are dependent, describing the same line in a two-dimensional plane. Consequently, there isn't just one solution; there's an entire infinite set of solutions, consisting of every point that lies on that line.

This outcome is significant because it represents a fundamental aspect of linear algebra – understanding the relationships between lines in a plane, and recognizing that not all systems will result in a single intersection point.