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In the world of algebra, a dependent system refers to a situation where the equations in a system are not distinct from each other. This means they essentially describe the same geometric object or line, rather than multiple ones intersecting. For example, in a two-variable scenario, this would translate to two lines that overlap completely, meaning they have every point in common.

In our exercise, the equations ultimately led to a statement like "0 = 0," which is always true. This indicates that no new information was gleaned from one equation over another. Hence, the entire system is dependent.

Understanding dependent systems is crucial as it helps us avoid unnecessary calculations. Instead of looking for a single solution, we aim to understand the conditions under which infinitely many solutions can exist.

A system with infinitely many solutions occurs when the equations in the system describe the same geometric entity in space, such as a line in two dimensions or a plane in three dimensions. This reflects the idea that there is not just one unique solution, but rather a whole set of solutions that satisfies all equations simultaneously.

In our system of equations:

• Equation 1: \(2a - b + c = 6\)
• Equation 2: \(-5a - 2b - 4c = -30\)
• Equation 3: \(a + b + c = 8\)

the dependent nature means that any combination of \(a\), \(b\), and \(c\) that satisfies one equation also satisfies the others. During the solution process, we ended up with a statement of equality \(0 = 0\), which affirmed that the equations were not providing new, unique constraints.

This is key in understanding that each solution is not just an arbitrary point, but part of a larger family of solutions sharing a common trait.

The technique of variable elimination is a strategic way to simplify solving a system of equations by reducing the number of unknowns systematically. This involves adjusting the equations strategically to cancel out one variable at a time, allowing you to focus on solving for the remaining variables.

In the provided exercise, we started by eliminating variable \(c\) from the first and third equations by expressing \(c\) in terms of \(a\) and \(b\). Then, this expression was substituted into other equations to reduce the system to fewer variables. This process simplifies the system to one that is more manageable and leads us to equations that more clearly reveal the relationship between the variables.

Key steps in variable elimination involve:

• Isolating one variable in one equation.
• Substituting it into the other equations.
• Simplifying and solving the resulting equations.

This technique is foundational in algebra, providing a stepping stone towards finding solutions or understanding when a system does not have a unique solution, as was the case in our dependent system with infinitely many solutions.