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A **system of equations** refers to a set of two or more equations with the same variables. The objective is to find the values of these variables that satisfy all the equations in the system simultaneously. In general, solving a system of equations involves finding the point or points where the graphs of the equations intersect.

• If the system has exactly one solution, the graphs intersect at a single point.
• If the system has no solution, the graphs do not intersect and are parallel.
• If the system has infinitely many solutions, the graphs are identical, and the equations are dependent.

Understanding how to solve a system of equations is a fundamental skill. It is widely applied in fields such as mathematics, physics, and engineering.

When two equations are dependent, they are graphically represented by **identical graphs**. This means the two equations describe the same line on a graph. When you plot both equations on a coordinate plane, you'll notice that the lines overlap perfectly.
Identical graphs imply that there is no unique solution to a system of equations, as each point on the line satisfies both equations. For instance, consider the equations given:

• Equation 1: \(2x + 3y = 6\)
• Equation 2: \(4x + 6y = 12\)

These equations have identical graphs, as Equation 2 is a multiple of Equation 1. Thus, every point on the line of Equation 1 is also on the line for Equation 2.

**Multiplying equations** involves scaling an entire equation by a non-zero constant. This operation is often used to identify whether two equations are dependent. By applying a multiplication operation to one equation, we can determine if another given equation can be obtained.
For example, in the exercise, the equation \(4x + 6y = 12\) is derived from multiplying each term in the equation \(2x + 3y = 6\) by 2. This illustrates the dependency between the two equations, as it was confirmed through multiplication that one equation can become the other.
When analyzing equations, if a system can be converted into identical equations through multiplication, it proves that the equations share the same graph and thus are dependent.

When equations are dependent, they inherently have the **same solutions**. Every input that satisfies one equation also satisfies the other. This is because dependent equations represent the same line in a graph.
So, for the given dependent equations \(2x + 3y = 6\) and \(4x + 6y = 12\), any combination of \(x\) and \(y\) that makes the first equation true will also make the second true, and vice versa. They do not only intersect at one point; they lay on top of each other completely, which reflects their dependency.
Understanding that dependent equations have the same solutions is crucial for recognizing identical graphs and the nature of the relationship between equations within a system.