Consistent And Inconsistent Systems – Introduction, Methods, Equations, Calculation & Solved Examples

# Consistent And Inconsistent Systems – Introduction, Methods, Equations, Calculation & Solved Examples

## Consistent Meaning In Maths

In mathematics, consistency is a property of sets of mathematical statements, meaning that no statement in the set contradicts any other. In other words, the statements in a consistent set of mathematical statements are all true together or all false together.

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## What does Inconsistent Systems Mean?

Inconsistent systems are those systems that are not logically consistent. This means that the system is not able to maintain a consistent state, and its internal logic is not able to support its own operations. This can lead to a number of problems, including data inconsistency, system crashes, and other errors.

## Difference Between Consistent and Inconsistent Systems

Inconsistent systems are those that cannot be relied upon to produce the same results each time they are used. For example, if you make a change to a document on your computer and save it, the change may not be saved the next time you open the document. This is an inconsistent system.

A consistent system, on the other hand, is one that always produces the same results. For example, a computer that is consistently reliable will save changes to a document each time it is opened.

## Consistent Meaning In Maths

In mathematics, a consistent meaning is a meaning that is always true, no matter what the circumstances. For example, the statement “two plus two equals four” is always true, no matter what.

## Dependent and Independent Systems

A system is said to be dependent if it cannot exist or function without another system. For example, the human body is a dependent system because it cannot survive without its organs.

A system is said to be independent if it can exist or function without another system. For example, a computer is an independent system because it can operate without input from the human body.

## Two-Variable Systems of Equations with Infinitely Many Solutions

A two-variable system of equations is a set of two equations in two variables. If a two-variable system of equations has infinitely many solutions, then it is said to be consistent.

An example of a two-variable system of equations with infinitely many solutions is the following:

\begin{align} 2x+y&=6\\ 3x+2y&=8\end{align}

The system above is consistent, and therefore has infinitely many solutions.

## The Elimination Method

The elimination method is a way to solve a system of linear equations. It is also known as the Gauss-Jordan Method.

To solve a system of linear equations using the elimination method, we perform the following steps:

1. Add the equations together to eliminate one of the variables.

2. Solve the resulting equation.

3. Substitute the result back into one of the original equations to solve for the other variable.

Let’s solve the following system of linear equations using the elimination method:

x + 2y = 3

3x – 2y = 5

Step 1: Add the equations together to eliminate one of the variables.

x + 2y = 3

3x – 2y = 5

x = 8

Step 2: Solve the resulting equation.

8 = 3

Step 3: Substitute the result back into one of the original equations to solve for the other variable.

x + 2y = 3

3(8) – 2y = 5

24 – 2y = 5

2y = 29

y = 14

### Example

Determine whether the given system of equations is inconsistent using either substitution or elimination. Provide a sketch of their graphs to supplement your answer.

Equation 1: x – 3y = 1

Equation 2: 2x – 8y = 3

Solving via Substitution

Step 1: To do this method, we choose one equation and substitute this to the other. In this case, we use equation 1 and rewrite it into

x = 3y + 1

Step 2: Now, we substitute equation 1 to equation 2 and solve for y.

2x – 8y = 3

2(3y + 1) – 8y = 3

6y + 2 – 8y = 3

(6 – 8)y = 3 – 2

y = – 1/2

Step 3: Then, we plug the value of y into either of the equations and solve for the value of x. Substituting this to equation 2 yields:

2x – 8y = 3

2x – 8(-1/2) = 3

2x + 4 = 3

2x = 3 – 4

x = -1/2

## A Consistent System of Equations

When it comes to the solution of a system of equations, there are three possible outcomes:

1. A finite number of solutions
2. Infinitely many solutions
3. No solution

Systems of equations can be placed into two categories: consistent and inconsistent. A consistent system of equations is a system that has at least one solution. An inconsistent system of equations is a system that has no solution. Thus, of the three possibilities for solutions of a system, we see that the first two possibilities represent consistent systems because they have at least one solution, and the third possibility represents an inconsistent system because it has no solution.

## Examples and Non-Examples

Consider the following system of equations:

4xy = 1

-8x + 2y = 4

Now,let’s examine the graph of this system:

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