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Consistent and Inconsistent Systems of Linear Equations

A system of linear equations is a set of equations with multiple variables that need to be solved simultaneously. These systems can be categorized as either consistent or inconsistent based on the existence of solutions. A consistent system has at least one set of values that satisfies all the equations in the system. In contrast, an inconsistent system has no solution because the equations contradict each other, such as when the lines are parallel and never intersect.

In this article, we will discuss Consistent and Inconsistent Systems of Linear Equations in detail.

Table of Content

Systems of Linear Equations

Systems of linear equations are a set of two or more linear equations involving the same set of variables.

These systems are foundational in mathematics, engineering, and various applied sciences because they model many real-world problems. The solutions to these systems are the points where the equations intersect in a graphical representation.

Types of Systems

These system of linear equations can be classified as:

  • Consistent Systems
    • Independent
    • Dependent
  • Inconsistent Systems

Consistent System of Linear Equations

A consistent system of linear equations is one that has at least one solution. Consistent systems can be classified into two categories:

  • Independent: The system has exactly one unique solution.
  • Dependent: The system has infinitely many solutions.

Solutions to Consistent Systems

A consistent system of linear equations can have either:

  • Unique Solution: This occurs when the system has exactly one solution. This is typically seen when the equations represent lines or planes that intersect at a single point.
  • Infinitely Many Solutions: This occurs when the system has more than one solution, often representing coincident lines or planes that overlap along a line or plane.

Examples of Consistent Systems

Consider the system of equations:

  • 2x + 3y = 5
  • 4x − y = 1

To solve this system, you can use methods such as substitution, elimination, or matrix operations. Solving this using substitution, we first solve the second equation for y:

  • y = 4x − 1

Substituting this into the first equation:

2x + 3(4x – 1) = 5

⇒ 2x + 12x – 3 = 5

⇒ 14x – 3 = 5

⇒ 14x = 8

⇒ x = 8/14 = 4/7

Using the value of x to find y:

y = 4(4/7) − 1 = 16/7 − 7/7 = 9/7

Thus, the solution to the system is x = 4/7​ and y = 9/7​, a unique solution.

Inconsistent System of Linear Equations

A system of linear equations is called inconsistent if it has no solutions.

An inconsistent system of linear equations is characterized by the fact that there is no point that satisfies all the equations simultaneously. This means that no matter what values are substituted for the variables, at least one equation will not be satisfied.

Implications of Inconsistent Systems

Graphically, an inconsistent system can be visualized as parallel lines that never intersect. For the example above, the lines represented by the equations x + y = 2 and x + y = 5 are parallel and distinct, indicating they have no points in common.

Examples of Inconsistent Systems

Consider the system of equations:

  • x + y = 2
  • x + y = 5

To see why this system is inconsistent, let’s analyze it:

  • The first equation states that the sum of x and y is 2.
  • The second equation states that the sum of x and y is 5.

It’s impossible for the same x and y values to satisfy both equations simultaneously. Therefore, the system has no solution.

How to Check Consistency of Linear Systems?

We can use the condition mentioned in the following table to check the consistency of systems of linear equations:

Method Condition for Consistency Description
Graphical Method Intersection If the graphs of the equations intersect at one point (consistent and independent) or overlap completely (consistent and dependent).
Algebraic Methods Solution Exists If solving the equations algebraically (substitution or elimination) yields a unique solution (consistent and independent) or a dependent solution (consistent and dependent).

Consistent Vs Inconsistent Systems

The key difference between consistent and inconsistent systems of linear equation are listed in the following table:

Criteria Consistent Systems Inconsistent Systems
Definition Systems with at least one solution Systems with no solutions
Number of Solutions One or infinitely many None
Graphical Representation Lines intersect at a point (one solution) or overlap (infinite solutions) Lines are parallel and never intersect
Equation Relationship Equations are dependent or intersect at a point Equations are independent and parallel
Example x + y = 2
x − y = 0
(solution: x = 1, y = 1)		 x + y = 2
x + y = 3
(no solution)
Augmented Matrix Form Has no row with all zeros except the last element Has a row with all zeros except the last element
Determinant (for square systems) Non-zero determinant (unique solution) or zero determinant (infinite solutions) Zero determinant (no solution)

Conclusion

In conclusion, consistent systems have at least one solution, where the lines representing the equations either intersect at a single point or overlap completely, indicating multiple solutions. In contrast, inconsistent systems have no solutions, as the lines are parallel and never meet. Recognizing these differences helps in determining the nature of the solutions and effectively solving the equations.

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FAQs on Consistent and Inconsistent Systems

Define consistent system of linear equations.

A consistent system of linear equations is a system that has at least one solution. This means the equations in the system intersect at one point or overlap entirely, resulting in a unique solution or infinitely many solutions.

What is an inconsistent system of linear equations?

An inconsistent system of linear equations is a system that has no solutions. This occurs when the equations represent parallel lines that never intersect.

How can you tell if a system of equations is consistent or inconsistent by looking at their graphs?

If the graphs of the equations intersect at a single point or overlap completely, the system is consistent. If the graphs are parallel and do not intersect, the system is inconsistent.

Can a consistent system have more than one solution?

Yes, a consistent system can have more than one solution. If the equations overlap completely (i.e., they are dependent), the system has infinitely many solutions.

What does the augmented matrix of a consistent system look like?

In the augmented matrix of a consistent system, there will be no row with all zeros except the last element. This indicates that there is at least one solution.

What does the augmented matrix of an inconsistent system look like?

In the augmented matrix of an inconsistent system, there will be at least one row where all elements are zero except the last one. This row indicates that the system has no solutions.


 


Reffered: https://www.geeksforgeeks.org

Mathematics

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