Consistent And Inconsistent Systems

Consistent Meaning In Maths

System of linear equations is a group of two or more linear equations having the same variables. For example, x + 2y = 14 , 2x + y  =   6.

To compare equations in linear systems, the best way is to see how many solutions both equations have in common. If there is nothing common between the two equations then it can be called as inconsistent. But it will be called consistent if any one ordered pair can solve both the equations. If the equation carries more than one point in common then it will be called as dependent. But what does ‘solution in common’ mean? It means that if there is at least one ordered pair which can solve both the equations in spite of having many equations that don’t.

For example, let us consider an equation  x + y = 6 and  x – y = 2. Do you think they have any solutions in common? Yes, Equation x + y = 6 does have many solutions but both of the equations have one solution in common i.e., if x = 4 and y=2 then both equations have true solutions. 


What Does Inconsistent Systems Mean?

Inconsistent equations of linear equations are equations that have no solutions in common. In this system, the lines will be parallel if the equations are graphed on a coordinate plane. Let's consider an inconsistent equations as  x – y = 8 and  5x – 5y = 25. They don’t have any common solutions. 

 When the lines or planes formed from the systems of equations don't meet at any point or are not parallel, it gives rise to an inconsistent system.


Consistent Meaning In Maths

consistent meaning in maths is an equation that has at least one solution in common. Let's take an example of consistent equations as x + y = 6 and x – y = 2 there is one solution in common. Similarly, in the equations x + y = 12 and 3y = x there is also one solution in common hence we can call them consistent equations.

If the lines formed by the equation meet at some point or are parallel then a two-variable system of equations to be considered consistent.

If a three-variable system of consistent linear equations is to be considered to be true then it must meet the following conditions:  

  1. All the three planes will have to parallel.

  2. Any two of the planes will have to be parallel. The third should meet one of the planes at some point while the other at another point.


Dependent And Independent Systems 

In a Dependent system, there are an infinite number of solutions that are in common and hence it is difficult to draw a single and unique solution. Graphically, both the equations can be graphed on the same line. Whereas in an independent system none of the equations can be derived from any other equations in the system.


Two variable system of equations with Infinitely Many Solutions

A two-variable system of equations is considered as equations of two lines and they can have infinitely many solutions if these two lines are parallel where they can be expressed as multiples of each other. This is a quick way to spot systems  with infinitely many solutions.


For example,

For the given equations, the variables can be solved using a substitution method. 

From equation (1) 

        

Substitute equation (2)

                     

It can be seen from the above equation that all the variables are lost which means that any value of x or y can be picked up. We can substitute it into any one of the two equations and therefore solve the other variable.

For example if we pick x = 0, then if we can substitute it into equation (1) to get y = 1. The value we pick up for x will always be different from the value of y. Thus, we can say that there are infinitely many solutions for the system of equations.


Two variable Systems of Equations with No Solution

Often we attempt to solve that system but end up with an equation that makes no sense mathematically because these equations are empty of any acceptable solution. Like for example, 

With the help of the matrix method we can solve the above equation as follows:

The reduction of the above matrix to Row Echelon form can be done as follows:

Now by Adding row 2 to row 1, we get:

The equation sketched out from the second row of the matrix is given as 

                                                             0x + 0y = 4

which means that:

0 = 4

We know that the above result is mathematically impossible and it can be said that the equation has no solution. Thus, such systems can be referred to as inconsistent as they don’t make any sense. 


The Elimination Method

In order to solve the variable in a system of equations, elimination method is used to eliminate the remaining variables. This elimination method is also known as elimination by addition. So, to find the correct value for the other variable it is substituted to the original equation after the values for the remaining variables are found.

Given Below are the Steps of the Elimination Method: 

  1. Line up the variables by rewriting the equations. 

  2. Modify one equation so that when the equations are added together,

  3.  both equations will have a variable that will cancel itself out.

  4. Equations need to be added and eliminate the variable.

  5. Solve the remaining variable.

  6. Solve for the other variable by back-substituting the previous one.

FAQ (Frequently Asked Questions)

Question 1) How do I Prove Consistent Linear Equations True?

Solution: In order to prove that a given system of linear equations is consistent, you must show that the ranks of the coefficient matrix as well as the corresponding augmented matrix associated with the given system are the same. The easiest way to establish this is to reduce the augmented matrix to a row-echelon form by using elementary row operations on it. It is to be noted that a homogeneous system of equations, i.e. in where the RHS of the equations are equal to 0, is always consistent.