How Do You Know if a System of Linear Equation Is Consistent

RS Aggarwal

In mathematics and particularly in algebra, a linear or nonlinear organisation of equations is called consequent if at that place is at to the lowest degree 1 gear up of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold truthful as an identity. In contrast, a linear or non linear equation organization is chosen inconsistent if there is no ready of values for the unknowns that satisfies all of the equations.^{[1] [two]}

If a system of equations is inconsistent, so it is possible to manipulate and combine the equations in such a way as to obtain contradictory data, such as two = 1, or x ³ + y ⁱⁱⁱ = 5 and 10 ³ + y ³ = 6 (which implies five = 6).

Both types of equation system, consistent and inconsistent, can exist whatever of overdetermined (having more than equations than unknowns), underdetermined (having fewer equations than unknowns), or exactly adamant.

Simple examples [edit]

Underdetermined and consistent [edit]

The organisation

${\displaystyle x+y+z=3,}$

${\displaystyle ten+y+2z=4}$

has an space number of solutions, all of them having z = i (as can be seen by subtracting the first equation from the second), and all of them therefore having x+y = two for any values of x and y.

The nonlinear system

${\displaystyle 10^{two}+y^{ii}+z^{2}=10,}$

${\displaystyle 10^{2}+y^{2}=5}$

has an infinitude of solutions, all involving ${\displaystyle z=\pm {\sqrt {5}}.}$

Since each of these systems has more than one solution, it is an indeterminate system.

Underdetermined and inconsistent [edit]

The system

${\displaystyle x+y+z=3,}$

${\displaystyle x+y+z=4}$

has no solutions, as tin be seen by subtracting the first equation from the second to obtain the impossible 0 = 1.

The non-linear system

${\displaystyle ten^{ii}+y^{2}+z^{2}=17,}$

${\displaystyle x^{2}+y^{2}+z^{2}=14}$

has no solutions, because if 1 equation is subtracted from the other nosotros obtain the impossible 0 = three.

Exactly determined and consequent [edit]

The system

${\displaystyle 10+y=iii,}$

${\displaystyle x+2y=v}$

has exactly one solution: x = 1, y = 2.

The nonlinear system

${\displaystyle x+y=1,}$

${\displaystyle x^{ii}+y^{two}=one}$

has the two solutions (x, y) = (i, 0) and (x, y) = (0, ane), while

${\displaystyle x^{3}+y^{3}+z^{3}=10,}$

${\displaystyle 10^{3}+2y^{3}+z^{3}=12,}$

${\displaystyle 3x^{iii}+5y^{3}+3z^{3}=34}$

has an space number of solutions because the third equation is the first equation plus twice the second ane and hence contains no independent data; thus any value of z tin be called and values of 10 and y tin be institute to satisfy the get-go two (and hence the tertiary) equations.

Exactly determined and inconsistent [edit]

The system

${\displaystyle x+y=iii,}$

${\displaystyle 4x+4y=10}$

has no solutions; the inconsistency tin can be seen by multiplying the first equation past 4 and subtracting the second equation to obtain the impossible 0 = two.

Likewise,

${\displaystyle x^{3}+y^{three}+z^{3}=ten,}$

${\displaystyle x^{3}+2y^{3}+z^{iii}=12,}$

${\displaystyle 3x^{3}+5y^{3}+3z^{3}=32}$

is an inconsistent system considering the first equation plus twice the second minus the third contains the contradiction 0 = two.

Overdetermined and consistent [edit]

The organisation

${\displaystyle x+y=three,}$

${\displaystyle x+2y=seven,}$

${\displaystyle 4x+6y=20}$

has a solution, x = –1, y = 4, because the kickoff two equations do not contradict each other and the third equation is redundant (since it contains the same information as can be obtained from the showtime two equations past multiplying each through by 2 and summing them).

The system

${\displaystyle x+2y=vii,}$

${\displaystyle 3x+6y=21,}$

${\displaystyle 7x+14y=49}$

has an infinitude of solutions since all three equations give the aforementioned data as each other (as tin be seen by multiplying through the first equation by either 3 or 7). Any value of y is part of a solution, with the corresponding value of x beingness 7–2y.

The nonlinear system

${\displaystyle x^{2}-1=0,}$

${\displaystyle y^{two}-1=0,}$

${\displaystyle (x-1)(y-1)=0}$

has the 3 solutions (10, y) = (1, –1), (–1, ane), and (1, one).

Overdetermined and inconsistent [edit]

The arrangement

${\displaystyle x+y=iii,}$

${\displaystyle x+2y=7,}$

${\displaystyle 4x+6y=21}$

is inconsistent because the last equation contradicts the information embedded in the first two, as seen by multiplying each of the first two through by ii and summing them.

The organisation

${\displaystyle x^{2}+y^{2}=one,}$

${\displaystyle 10^{2}+2y^{ii}=2,}$

${\displaystyle 2x^{2}+3y^{2}=four}$

is inconsistent because the sum of the first two equations contradicts the third ane.

Criteria for consistency [edit]

Every bit can be seen from the in a higher place examples, consistency versus inconsistency is a unlike event from comparing the numbers of equations and unknowns.

Linear systems [edit]

A linear system is consistent if and just if its coefficient matrix has the same rank as does its augmented matrix (the coefficient matrix with an extra column added, that column being the column vector of constants).

Nonlinear systems [edit]

References [edit]

1. ^ "Definition of Consequent EQUATIONS". www.merriam-webster.com . Retrieved 2021-06-10 .
2. ^ "Definition of consistent equations | Dictionary.com". world wide web.dictionary.com . Retrieved 2021-06-10 .

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Source: https://en.wikipedia.org/wiki/Consistent_and_inconsistent_equations

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