[Maths Class Notes] on Consistent And Inconsistent Systems Pdf for Exam

A 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 inconsistent. But it will be called consistent if anyone ordered pair can solve both the equations. If the equation carries more than one point in common then it will be called dependent. But what does ‘solution in common’ mean? It means that if there is at least one ordered pair that can solve both the equations in spite of having many equations that do not.

A system of equations is formed by the two equations y=2x+5 and y=4x+3. The system’s solution is the ordered pair that is the solution of both equations.

There can be a single solution, an infinite number of solutions, or no solution to a system of two linear equations. The number of solutions in a system of equations can be used to differentiate it.

A system is said to be consistent if it has a minimum of one solution.

It is independent if a consistent system has only one solution.

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 equation 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.

Difference Between Consistent and Inconsistent Systems

A linear or nonlinear system of equations is considered to be consistent in mathematics and especially algebra if at least one set of values for the unknowns satisfies each equation in the system—that is, when substituted into every equation, they make each equation turn true as an identity. The term inconsistent is utilized to delineate a linear or nonlinear equation system in which no set of values for the unknown fulfills all of the equations.

Consistent Meaning In Maths

A 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 is 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 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 Systems 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.

The Elimination Method

In order to solve the variable in a system of equations, an 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 variables.

5. Solve the remaining variable.

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