Classification of Systems of Linear Equations

Oct 13, 2025 by magicalceylontours.com 12 views

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. These equations are linear because they can be written in the form of a linear equation, where the variables are raised to the power of 1. The system of linear equations can be classified into three categories: consistent, inconsistent, and dependent. In this article, we will discuss the classification of the given system of linear equations and provide the solution to each system.

System 1: Consistent or Inconsistent?

The first system of linear equations is given by:

${ \begin{array}{l} 3x - 4y = 12 \\ 6x - 8y = 21 \end{array} }$

To determine whether this system is consistent or inconsistent, we need to check if the two equations are parallel or not. If the two equations are parallel, then the system is inconsistent. Otherwise, the system is consistent.

To check if the two equations are parallel, we can divide the second equation by 2. This gives us:

${ \begin{array}{l} 3x - 4y = 12 \\ 3x - 4y = 10.5 \end{array} }$

As we can see, the two equations are not equal, and therefore, they are not parallel. This means that the system is consistent.

System 2: Consistent or Inconsistent?

The second system of linear equations is given by:

${ \begin{array}{l} 2x - 3y = 8 \\ -3x + 2y = 8 \end{array} }$

To check if the two equations are parallel, we can multiply the first equation by 3 and the second equation by 2. This gives us:

${ \begin{array}{l} 6x - 9y = 24 \\ -6x + 4y = 16 \end{array} }$

As we can see, the two equations are not equal, and therefore, they are not parallel. This means that the system is consistent.

Solution to System 1

Since the system is consistent, we can solve for the values of x and y. To do this, we can use the method of substitution or elimination.

Let's use the elimination method. We can multiply the first equation by 2 and the second equation by 1. This gives us:

${ \begin{array}{l} 6x - 8y = 24 \\ 6x - 8y = 21 \end{array} }$

As we can see, the two equations are equal, and therefore, we can subtract one equation from the other to eliminate the variable x. This gives us:

${ 0 = 3 }$

This is a contradiction, and therefore, the system has no solution.

Solution to System 2

Since the system is consistent, we can solve for the values of x and y. To do this, we can use the method of substitution or elimination.

Let's use the elimination method. We can multiply the first equation by 2 and the second equation by 1. This gives us:

${ \begin{array}{l} 4x - 6y = 16 \\ -3x + 2y = 8 \end{array} }$

As we can see, the two equations are not equal, and therefore, we cannot eliminate the variable x. However, we can add the two equations to eliminate the variable y. This gives us:

${ x = 12 }$

Now that we have the value of x, we can substitute it into one of the original equations to solve for the value of y. Let's use the first equation:

${ 2x - 3y = 8 }$

Substituting x = 12, we get:

${ 24 - 3y = 8 }$

Solving for y, we get:

${ y = 4 }$

Therefore, the solution to the system is x = 12 and y = 4.

Conclusion

In this article, we discussed the classification of the given system of linear equations and provided the solution to each system. We showed that the first system is inconsistent, and the second system is consistent. We also provided the solution to the second system, which is x = 12 and y = 4.

Classification of Systems of Linear Equations

A system of linear equations can be classified into three categories: consistent, inconsistent, and dependent.

Consistent System: A consistent system is one that has a solution. In other words, the system has at least one solution.
Inconsistent System: An inconsistent system is one that has no solution. In other words, the system has no solution.
Dependent System: A dependent system is one that has an infinite number of solutions. In other words, the system has an infinite number of solutions.

Method of Substitution

The method of substitution is a technique used to solve a system of linear equations. This method involves substituting the value of one variable into the other equation to solve for the value of the other variable.

Method of Elimination

The method of elimination is a technique used to solve a system of linear equations. This method involves eliminating one variable by adding or subtracting the two equations.

Conclusion

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. These equations are linear because they can be written in the form of a linear equation, where the variables are raised to the power of 1.

Q: How do I determine if a system of linear equations is consistent or inconsistent?

A: To determine if a system of linear equations is consistent or inconsistent, you need to check if the two equations are parallel or not. If the two equations are parallel, then the system is inconsistent. Otherwise, the system is consistent.

Q: What is the difference between a consistent and an inconsistent system?

A: A consistent system is one that has a solution, while an inconsistent system is one that has no solution.

Q: How do I solve a system of linear equations?

A: There are two main methods to solve a system of linear equations: the method of substitution and the method of elimination. The method of substitution involves substituting the value of one variable into the other equation to solve for the value of the other variable. The method of elimination involves eliminating one variable by adding or subtracting the two equations.

Q: What is the method of substitution?

A: The method of substitution is a technique used to solve a system of linear equations. This method involves substituting the value of one variable into the other equation to solve for the value of the other variable.

Q: What is the method of elimination?

A: The method of elimination is a technique used to solve a system of linear equations. This method involves eliminating one variable by adding or subtracting the two equations.

Q: Can a system of linear equations have an infinite number of solutions?

A: Yes, a system of linear equations can have an infinite number of solutions. This is known as a dependent system.

Q: What is a dependent system?

A: A dependent system is a system of linear equations that has an infinite number of solutions.

Q: How do I determine if a system of linear equations is dependent or independent?

A: To determine if a system of linear equations is dependent or independent, you need to check if the two equations are parallel or not. If the two equations are parallel, then the system is dependent. Otherwise, the system is independent.

Q: What is the difference between a dependent and an independent system?

A: A dependent system is one that has an infinite number of solutions, while an independent system is one that has a unique solution.

Q: Can a system of linear equations have a unique solution?

A: Yes, a system of linear equations can have a unique solution. This is known as an independent system.

Q: How do I determine if a system of linear equations has a unique solution?

A: To determine if a system of linear equations has a unique solution, you need to check if the two equations are not parallel. If the two equations are not parallel, then the system has a unique solution.

Conclusion

In conclusion, the system of linear equations can be classified into three categories: consistent, inconsistent, and dependent. We discussed the classification of the given system of linear equations and provided the solution to each system. We also answered some frequently asked questions about systems of linear equations.