Consistent and inconsistent equations

In mathematics and particularly in algebra, a system of equations (either linear or nonlinear) is called consistent if there is at least one set 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 true as an identity. In contrast, a linear or non linear equation system is called inconsistent if there is no set of values for the unknowns that satisfies all of the equations.^[1]^[2]

If a system of equations is inconsistent, then the equations cannot be true together leading to contradictory information, such as the false statements $2 = 1$ , or $x^{3}+y^{3}=5$ and $x^{3}+y^{3}=6$ (which implies $5 = 6$ ).

Both types of equation system, consistent and inconsistent, can be any of overdetermined (having more equations than unknowns), underdetermined (having fewer equations than unknowns), or exactly determined.

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Is the system of linear equations below consistent or inconsistent? And they give us x plus 2y is equal to 13 and 3x minus y is equal to negative 11. So to answer this question, we need to know what it means to be consistent or inconsistent. So a consistent system of equations. has at least one solution. And an inconsistent system of equations, as you can imagine, has no solutions. So if we think about it graphically, what would the graph of a consistent system look like? Let me just draw a really rough graph. So that's my x-axis, and that is my y-axis. So if I have just two different lines that intersect, that would be consistent. So that's one line, and then that's another line. They clearly have that one solution where they both intersect, so that would be a consistent system. Another consistent system would be if they're the same line, because then they would intersect at a ton of points, actually at an infinite number of points. So let's say one of the lines looks like that. And then the other line is actually the exact same line. So it's exactly right on top of it. So those two intersect at every point along those lines, so that also would be consistent. An inconsistent system would have no solutions. So let me again draw my axes. Let me once again draw my axes. It will have no solutions. And so the only way that you're going to have two lines in two dimensions have no solutions is if they don't intersect, or if they are parallel. So one line could look like this. And then the other line would have the same slope, but it would be shifted over. It would have a different y-intercept, so it would look like this. So that's what an inconsistent system would look like. You have parallel lines. This right here is inconsistent. So what we could do is just do a rough graph of both of these lines and see if they intersect. Another way to do it is, you could look at the slope. And if they have the same slope and different y-intercepts, then you'd also have an inconsistent system. But let's just graph them. So let me draw my x-axis and let me draw my y-axis. So this is x and then this is y. And then there's a couple of ways we could do it. The easiest way is really just find two points on each of these that satisfy each of these equations, and that's enough to define a line. So for this first one, let's just make a little table of x's and y's. When x is 0, you have 2y is equal to 13, or y is equal to 13/2, which is the same thing as 6 and 1/2. So when x is 0, y is 6 and 1/2. I'll just put it right over here. So this is 0 comma 13/2. And then let's just see what happens when y is 0. When y is 0, then 2 times y is 0. You have x equaling 13. x equals 13. So we have the point 13 comma 0. So this is 0, 6 and 1/2, so 13 comma 0 would be right about there. We're just trying to approximate-- 13 comma 0. And so this line right up here, this equation can be represented by this line. Let me try my best to draw it. It would look something like that. Now let's worry about this one. Let's worry about that one. So once again, let's make a little table, x's and y's. I'm really just looking for two points on this graph. So when x is equal to 0, 3 times 0 is just 0. So you get negative y is equal to negative 11, or you get y is equal to 11. So you have the point 0, 11, so that's maybe right over there. 0 comma 11 is on that line. And then when y is 0, you have 3x minus 0 is equal to negative 11, or 3x is equal to negative 11. Or if you divide both sides by 3, you get x is equal to negative 11/3. And this is the exact same thing as negative 3 and 2/3. So when y is 0, you have x being negative 3 and 2/3. So maybe this is about 6, so negative 3 and 2/3 would be right about here. So this is the point negative 11/3 comma 0. And so the second equation will look like something like this. Will look something like that. Now clearly-- and I might have not been completely precise when I did this hand-drawn graph-- clearly these two guys intersect. They intersect right over here. And to answer their question, you don't even have to find the point that they intersect at. We just have to see, very clearly, that these two lines intersect. So this is a consistent system of equations. It has one solution. You just have to have at least one in order to be consistent. So once again, consistent system of equations.

Simple examples

Underdetermined and consistent

The system

{\begin{aligned}x+y+z&=3,\\x+y+2z&=4\end{aligned}}

has an infinite number of solutions, all of them having $z = 1$ (as can be seen by subtracting the first equation from the second), and all of them therefore having $x + y = 2$ for any values of $x$ and $y$ .

The nonlinear system

{\begin{aligned}x^{2}+y^{2}+z^{2}&=10,\\x^{2}+y^{2}&=5\end{aligned}}

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

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

Underdetermined and inconsistent

The system

{\begin{aligned}x+y+z&=3,\\x+y+z&=4\end{aligned}}

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

The non-linear system

{\begin{aligned}x^{2}+y^{2}+z^{2}&=17,\\x^{2}+y^{2}+z^{2}&=14\end{aligned}}

has no solutions, because if one equation is subtracted from the other we obtain the impossible $0 = 3$ .

Exactly determined and consistent

The system

{\begin{aligned}x+y&=3,\\x+2y&=5\end{aligned}}

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

The nonlinear system

{\begin{aligned}x+y&=1,\\x^{2}+y^{2}&=1\end{aligned}}

has the two solutions $(x, y) = (1, 0)$ and $(x, y) = (0, 1)$ , while

{\begin{aligned}x^{3}+y^{3}+z^{3}&=10,\\x^{3}+2y^{3}+z^{3}&=12,\\3x^{3}+5y^{3}+3z^{3}&=34\end{aligned}}

has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of $z$ can be chosen and values of $x$ and $y$ can be found to satisfy the first two (and hence the third) equations.

Exactly determined and inconsistent

The system

{\begin{aligned}x+y&=3,\\4x+4y&=10\end{aligned}}

has no solutions; the inconsistency can be seen by multiplying the first equation by 4 and subtracting the second equation to obtain the impossible $0 = 2$ .

Likewise,

{\begin{aligned}x^{3}+y^{3}+z^{3}&=10,\\x^{3}+2y^{3}+z^{3}&=12,\\3x^{3}+5y^{3}+3z^{3}&=32\end{aligned}}

is an inconsistent system because the first equation plus twice the second minus the third contains the contradiction $0 = 2$ .

Overdetermined and consistent

The system

{\begin{aligned}x+y&=3,\\x+2y&=7,\\4x+6y&=20\end{aligned}}

has a solution, $x = -1, y = 4$ , because the first 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 first two equations by multiplying each through by 2 and summing them).

The system

{\begin{aligned}x+2y&=7,\\3x+6y&=21,\\7x+14y&=49\end{aligned}}

has an infinitude of solutions since all three equations give the same information as each other (as can 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 $x$ being $7 - 2 y$ .

The nonlinear system

{\begin{aligned}x^{2}-1&=0,\\y^{2}-1&=0,\\(x-1)(y-1)&=0\end{aligned}}

has the three solutions $(x, y) = (1, -1), (-1, 1), (1, 1)$ .

Overdetermined and inconsistent

The system

{\begin{aligned}x+y&=3,\\x+2y&=7,\\4x+6y&=21\end{aligned}}

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 2 and summing them.

The system

{\begin{aligned}x^{2}+y^{2}&=1,\\x^{2}+2y^{2}&=2,\\2x^{2}+3y^{2}&=4\end{aligned}}

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

Criteria for consistency

As can be seen from the above examples, consistency versus inconsistency is a different issue from comparing the numbers of equations and unknowns.

Linear systems

A linear system is consistent if and only 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

References

^ "Definition of CONSISTENT EQUATIONS". www.merriam-webster.com. Retrieved 2021-06-10.
^ "Definition of consistent equations | Dictionary.com". www.dictionary.com. Retrieved 2021-06-10.

This page was last edited on 8 April 2024, at 10:29

From Wikipedia, the free encyclopedia

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Transcription

Simple examples

Underdetermined and consistent

Underdetermined and inconsistent

Exactly determined and consistent

Exactly determined and inconsistent

Overdetermined and consistent

Overdetermined and inconsistent

Criteria for consistency

Linear systems

Nonlinear systems

References