What does it mean for a system of equations to be consistent?

Looking at a system of equations with only one solution? That means that those equations intersect only at that one point. That kind of solution is called consistent and independent! This tutorial explains systems with one solution and even shows you an example!

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.

To sketch the graph of pair of linear equations in two variables, we draw two lines representing the equations. The following cases are possible:

i) If both the lines intersect at a point, then there exists a unique solution to the pair of linear equations. In such a case, the pair of linear equations is said to be consistent.

In the graph given above, lines intersect at point P(x, y)  which represents the unique solution of the system of linear equations in two variables.

\(\begin{array}{l}Algebraically,\ if\ \frac{a_1}{a_2}~ \neq ~ \frac{b_1}{b_2} \ then,\ the\ linear\ equation\ pair\ is\ consistent.\end{array} \)

ii) Consider two lines having equation to be-

\(\begin{array}{l}a_1 x + b_1 y + c_1 = 0 \ and \ a_2 x + b_2 y + c_2 = 0\end{array} \)

Let these lines coincide with each other, then there exist infinitely many solutions since a line consists of infinite points. In such a case, the pair of linear equations is said to be dependent and consistent. As represented in the graph below, the pair of lines coincide and, therefore, dependent and consistent.

Algebraically, when a1/a2 = b1/b2 = c1/c2,  then the lines coincide and the pair of equations is dependent and consistent.

Read more:

• Cramer’s rule
• Simultaneous linear equations
• Linear equations

Inconsistent System

i) Consider the equation of the lines to be-

\(\begin{array}{l}a_1 x + b_1 y + c_1 = 0 \ and \ a_2 x + b_2 y + c_2 = 0\end{array} \)

Let both the lines to be parallel to each other, then there exists no solution because the lines never intersect.

Algebraically, for such a case, a1/a2 = b1/b2 ≠ c1/c2, and the pair of linear equations in two variables is said to be inconsistent.

As shown in the graph above, the pair of lines  a1x +b1y +c1 =0 and a2x +b2y +c1 =0 are parallel to each other.

Therefore, there exists no solution for such a pair.

Solved Example

Check for the consistency of the following pair of linear equations-

\(\begin{array}{l}x – 2y = 1\end{array} \)

\(\begin{array}{l}2x – 4y=2\end{array} \)

Solution:

To check the condition of consistency we need to find out the ratios of the coefficients of the given equations,

\(\begin{array}{l}\frac{a_1}{a_2} = \frac{1}{2}\end{array} \)

\(\begin{array}{l}\frac{b_1}{b_2} = \frac{1}{2}\end{array} \)

\(\begin{array}{l}\frac{c_1}{c_2} = \frac{1}{2}\end{array} \)

Thus,

\(\begin{array}{l}\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\end{array} \)

So, we can say that the above equations represent lines which are coincident in nature and the pair of equations is dependent and consistent.

Therefore, we can say that the lines coincide with each other, having an infinite number of solutions.

Also, when we plot the given equations on a graph, it represents a pair of coincident lines.

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