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When a system of linear equations has infinite solutions, it means that there are countless solutions available. This typically occurs when the equations actually represent the same line. If you plot the equations on a graph, you will find that both lines lie directly on top of each other.

In the exercise given, the second equation is simply a multiple of the first. This is a clear indicator of infinite solutions, as it suggests they are just different expressions of the same line. So, no matter what value you plug in for one of the variables (like \(x\)), you can always find a corresponding \(y\) that satisfies both equations.

To summarize, infinite solutions arise when the lines, represented by the equations, do not just intersect at a single point, but overlap completely. Therefore, any point on the line is a solution to the system.

A system of equations is termed consistent when there is at least one solution. It becomes dependent when it has infinite solutions, meaning the equations describe the same line.

In our exercise, since both equations represent the same line, the system is consistent (because solutions exist) and dependent (because it has infinite solutions). This happens when one equation can be derived from another by multiplying by a constant.

Features of a consistent and dependent system include:

• All solutions satisfy every equation in the system.
• The graphs of the equations are identical.
• The system is always consistent as long as the equations do not contradict each other.

A useful tip to spot if a system is consistent and dependent is to check if every equation is proportionally equal or relates to the others by a set multiplier.

Analyzing systems of linear equations is crucial for finding whether they have one solution, no solution, or infinite solutions. Here's a quick rundown on how to analyze systems:

• Check the coefficients of the variables. If multiplying one equation by a number makes it equal to another, you have infinite solutions.
• If the equations intersect at exactly one point, the system has a unique solution.
• If the lines are parallel (same slope but different intercepts), there are no solutions.

In our example, by examining the coefficients, we see the second equation is double the first. This fact tells us the system has infinite solutions.

Understanding how to analyze equations allows you to quickly identify relationships between them, greatly simplifying the process of finding solutions.