Which System Of Equations Is Consistent And Dependent?A. ${ \left{ \begin{array}{l} 3x + 2y = 3 \ 6x + 4y = 6 \end{array} \right. }$B. ${ \left{ \begin{array}{l} x + 8y = 2 \ 4x + 6y = 1 \end{array} \right. }$C. $[
Introduction
In mathematics, a system of equations is a set of equations that involve multiple variables. These equations can be linear or non-linear, and they can be used to model a wide range of real-world problems. When solving a system of equations, we often encounter two types of systems: consistent and inconsistent systems, and dependent and independent systems. In this article, we will focus on identifying which system of equations is consistent and dependent.
What are Consistent and Dependent Systems?
A consistent system of equations is one that has at least one solution. In other words, there exists a set of values for the variables that satisfy all the equations in the system. On the other hand, an inconsistent system of equations is one that has no solution. This means that there is no set of values for the variables that can satisfy all the equations in the system.
A dependent system of equations is one in which the equations are not independent of each other. In other words, one equation can be expressed as a multiple of the other equation. This means that the equations are not distinct and do not provide any new information.
**System A: ${ \left{ \begin{array}{l} 3x + 2y = 3 \ 6x + 4y = 6 \end{array} \right. }$
Let's start by analyzing System A. We can see that the two equations are:
We can simplify the second equation by dividing both sides by 2, which gives us:
This shows that the two equations are identical, and therefore, the system is dependent. Since the system is dependent, it is also consistent, as there are an infinite number of solutions.
**System B: ${ \left{ \begin{array}{l} x + 8y = 2 \ 4x + 6y = 1 \end{array} \right. }$
Now, let's analyze System B. We can see that the two equations are:
We can multiply the first equation by 4, which gives us:
Now, we can subtract the second equation from this new equation, which gives us:
This shows that the system has a unique solution, and therefore, it is consistent and independent.
**System C: ${ [ \left{ \begin{array}{l} x + 2y = 1 \ x + 2y = 2 \end{array} \right. }$
Finally, let's analyze System C. We can see that the two equations are:
We can subtract the first equation from the second equation, which gives us:
This shows that the system is inconsistent, as there is no solution.
Conclusion
In conclusion, we have analyzed three systems of equations and determined which one is consistent and dependent. System A is consistent and dependent, as the two equations are identical and there are an infinite number of solutions. System B is consistent and independent, as the two equations have a unique solution. System C is inconsistent, as there is no solution.
Key Takeaways
- A consistent system of equations is one that has at least one solution.
- A dependent system of equations is one in which the equations are not independent of each other.
- System A is consistent and dependent, as the two equations are identical and there are an infinite number of solutions.
- System B is consistent and independent, as the two equations have a unique solution.
- System C is inconsistent, as there is no solution.
Final Thoughts
Introduction
In our previous article, we discussed the concept of consistent and dependent systems of equations. We analyzed three systems of equations and determined which one is consistent and dependent. In this article, we will provide a Q&A section to help clarify any doubts and provide further understanding of the concept.
Q: What is the difference between a consistent and an inconsistent system of equations?
A: A consistent system of equations is one that has at least one solution. In other words, there exists a set of values for the variables that satisfy all the equations in the system. On the other hand, an inconsistent system of equations is one that has no solution. This means that there is no set of values for the variables that can satisfy all the equations in the system.
Q: What is the difference between a dependent and an independent system of equations?
A: A dependent system of equations is one in which the equations are not independent of each other. In other words, one equation can be expressed as a multiple of the other equation. This means that the equations are not distinct and do not provide any new information. An independent system of equations, on the other hand, is one in which the equations are distinct and provide new information.
Q: How can I determine if a system of equations is consistent and dependent?
A: To determine if a system of equations is consistent and dependent, you can follow these steps:
- Write down the system of equations.
- Simplify the equations by combining like terms.
- Check if the equations are identical. If they are, then the system is dependent.
- Check if the system has a unique solution. If it does, then the system is consistent and independent.
Q: What is an example of a consistent and dependent system of equations?
A: An example of a consistent and dependent system of equations is:
{ \left\{ \begin{array}{l} 3x + 2y = 3 \\ 6x + 4y = 6 \end{array} \right. \}
In this system, the two equations are identical, and therefore, the system is dependent. Since the system is dependent, it is also consistent, as there are an infinite number of solutions.
Q: What is an example of a consistent and independent system of equations?
A: An example of a consistent and independent system of equations is:
{ \left\{ \begin{array}{l} x + 8y = 2 \\ 4x + 6y = 1 \end{array} \right. \}
In this system, the two equations are distinct, and therefore, the system is independent. Since the system is independent, it is also consistent, as there is a unique solution.
Q: What is an example of an inconsistent system of equations?
A: An example of an inconsistent system of equations is:
{ \left\{ \begin{array}{l} x + 2y = 1 \\ x + 2y = 2 \end{array} \right. \}
In this system, the two equations are identical, and therefore, the system is dependent. However, since the system is dependent, it is also inconsistent, as there is no solution.
Conclusion
In conclusion, we have provided a Q&A section to help clarify any doubts and provide further understanding of the concept of consistent and dependent systems of equations. We hope that this article has been helpful in providing a clear understanding of the concept.
Key Takeaways
- A consistent system of equations is one that has at least one solution.
- A dependent system of equations is one in which the equations are not independent of each other.
- System A is consistent and dependent, as the two equations are identical and there are an infinite number of solutions.
- System B is consistent and independent, as the two equations have a unique solution.
- System C is inconsistent, as there is no solution.
Final Thoughts
In this article, we have provided a Q&A section to help clarify any doubts and provide further understanding of the concept of consistent and dependent systems of equations. We hope that this article has been helpful in providing a clear understanding of the concept.