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Consider the equations:

$ x+2y + 2z =1$

and, $ 2x+4y+4z=9$

According to my book, we can infer there are zero solutions from looking at both equations. However, this doesn't make sense to me as, I was taught that we find out about number of solution using the determinant of the coefficients of three equations for a linear system of three variables.

Basically my question is, given two equation of three variables, can we infer about the number of solutions without having a third?

tryst with freedom
  • 11,538
  • $(x,y,z)=(1,0,0)$ is a solution over the field $\Bbb F_7$. – Dietrich Burde May 28 '20 at 11:34
  • what is a field – tryst with freedom May 28 '20 at 11:36
  • For example, $\Bbb Q$, or $\Bbb R$, or $\Bbb C$, or $\Bbb F_p$. See wikipedia. Where did you want to have the solutions? In the integers? – Dietrich Burde May 28 '20 at 11:37
  • Yes, you can use the determinant of the coefficient matrix when you have enough equations, but that’s not applicable here since there are fewer equations than unknowns. The determinant by itself doesn’t really tell you enough even when there are. If it vanishes, then you need to examine other things to determine whether there are zero or an infinite number of solutions to the system. – amd May 30 '20 at 01:44

1 Answers1

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You may simply observe that once multiplied by 2 the first equation becomes $2x+4y+4z=2$. However since $2\neq 9$ we may note that for all values these two equations are inconsistent and so there are no solutions.

  • I meant to say don't we need 4 equations to determine consistency – tryst with freedom May 28 '20 at 11:26
  • No. And btw: the question whether that makes sense to you won't make much sense to the universe. –  May 28 '20 at 11:26
  • In general for $n$ distinct equations in $n$ variables you can determine the a solution through the method you stated above. However in general given two equations you cannot say for certain without observing properties of the functions. For example in the case you gave you can observe the two planes run parallel and so there are no solutions. Further they could be the same equation and so you can have infinitely many solutions in that case also, provided by the plane. Finally they could intersect in a line so you obtain infinitely many solutions again there. In general no. –  May 28 '20 at 11:30
  • I meant '3' equations sorry – tryst with freedom May 28 '20 at 11:35
  • So, what exactly is the algorithim for determining if there is solutions or not? – tryst with freedom May 28 '20 at 11:38
  • @DDD4C4U I assumed that so answered with considering the case of two planes (e.g. two equations given in 3 variables) and considered the possibilities in how they can interact. The number of solutions will vary by the consideration of another plane. If the three planes are distinct they will intersect at one point (can you see why?) and otherwise if you have two planes we are in the case above. Does this clarify your issue? If not can you try to be more specific and I'll see if I can explain it further. –  May 28 '20 at 11:39
  • the thing which threw me off in the first comment is the stuff about functions. Other than that I think I get it now, how do u check if two planes are distinct or not? – tryst with freedom May 28 '20 at 11:43
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    By that with functions I meant in general just consider the coefficients. Essentially the planes are the same if you can reduce the coefficients down to the simplest possible form and if they are the same consider the constant term. Generally if the coefficients are different then the equations are different. You just have to be careful in the case say :$x+y+z=1$ and $-x-y-z=-1$ since if you multiply the second plane by -1 you have the same equation and thus same plane. –  May 28 '20 at 11:46
  • I didn't get a notification of this yesterday but I understood now :) – tryst with freedom May 30 '20 at 08:34