Consistent And Independent System Of Equations Explained
When faced with a system of equations, determining the nature of its solutions is crucial. This involves classifying the system based on its consistency and dependence. In this article, we will delve deep into understanding these concepts and apply them to the given system of equations:
$ \begin{array}{l} 3 y=9 x-6 \ 2 y+6 x=4 \end{array} $
By carefully analyzing the equations, we can identify whether they are consistent or inconsistent and whether they are dependent or independent. This analysis will lead us to the correct classification of the system.
Understanding Consistent and Inconsistent Systems
In the realm of systems of equations, consistency refers to the existence of a solution. A consistent system is one that has at least one solution, meaning there is at least one set of values for the variables that satisfies all equations in the system simultaneously. On the other hand, an inconsistent system is one that has no solution. This means there is no set of values for the variables that can satisfy all equations in the system at the same time. The lines represented by the equations in an inconsistent system are parallel and never intersect.
To determine the consistency of a system, we can manipulate the equations to see if they lead to a contradiction. A contradiction arises when, through algebraic manipulations, we arrive at a statement that is always false, such as 0 = 1. If we encounter a contradiction, the system is inconsistent. If no contradiction arises, the system is consistent.
Consider the system:
$ \begin{cases} x + y = 5 \ x + y = 10 \end{cases} $
This system is inconsistent because it is impossible for two numbers to add up to both 5 and 10 simultaneously. If we were to subtract the first equation from the second, we would get 0 = 5, a clear contradiction.
Conversely, the system:
$ \begin{cases} x + y = 5 \ x - y = 1 \end{cases} $
is consistent because it has a solution (x = 3, y = 2). These values satisfy both equations.
Exploring Dependent and Independent Systems
While consistency deals with the existence of solutions, dependence describes the relationship between the equations within a system. A dependent system is one where the equations are essentially the same, meaning one equation is a multiple of the other. In graphical terms, dependent equations represent the same line. Consequently, a dependent system has infinitely many solutions, as any point on the line satisfies both equations. An independent system, conversely, has equations that are not multiples of each other. These equations represent distinct lines that intersect at a single point, resulting in a unique solution.
To identify dependence, we can try to manipulate one equation to match the other. If we can do so, the equations are dependent. If not, they are independent.
For example, the system:
$ \begin{cases} 2x + 4y = 6 \ x + 2y = 3 \end{cases} $
is dependent because the first equation is simply twice the second equation. Both equations represent the same line.
However, the system:
$ \begin{cases} x + y = 5 \ x - y = 1 \end{cases} $
is independent because the equations are not multiples of each other and represent distinct lines.
Analyzing the Given System
Now, let's apply these concepts to the given system of equations:
$ \begin{array}{l} 3 y=9 x-6 \ 2 y+6 x=4 \end{array} $
First, we can rewrite the equations in slope-intercept form (y = mx + b) to better understand their relationship.
The first equation, 3y = 9x - 6, can be rewritten as:
The second equation, 2y + 6x = 4, can be rewritten as:
Now we have the system in slope-intercept form:
$ \begin{cases} y = 3x - 2 \ y = -3x + 2 \end{cases} $
By observing the equations in slope-intercept form, we can analyze their slopes and y-intercepts.
Determining Consistency
The slopes of the two lines are 3 and -3, respectively. Since the slopes are different, the lines are not parallel and will intersect at a single point. This indicates that the system has a solution, making it consistent.
Determining Dependence
The equations are not multiples of each other. There is no constant that we can multiply one equation by to obtain the other. This means the equations are independent.
Conclusion: Consistent and Independent
Based on our analysis, the given system of equations is consistent because it has a solution and independent because the equations are not multiples of each other. Therefore, the correct answer is A. consistent and independent.
Understanding the concepts of consistency and dependence is vital for solving and interpreting systems of equations. By analyzing the equations and their relationships, we can effectively classify the system and determine the nature of its solutions. In this case, the system's consistency and independence point to a unique solution, which can be found through methods like substitution or elimination.
This exploration of consistent and inconsistent, dependent and independent systems provides a solid foundation for tackling more complex mathematical problems involving multiple equations and variables. The ability to classify systems accurately is a key skill in algebra and beyond.
