How can you identify how many solutions systems have in $\mathbb{R}^3$?

Question

How to determine how many solutions does each of the following systems have in $\mathbb{R}^3$? (Infinitely many solutions/Unique solutions/No Solutions)

a) $\begin{cases}f + g + h = 13\\ f – h = −2\end{cases}$

b) $\begin{cases}3x + 4y – z = 8\\ 5x – 2y + z = 4\\ 2x – 2y + z = 1\end{cases}$

c) $\begin{cases}S – T – W = 8\\ 5S + 2T + 4W = 0\\ –3S + 3T + 3W =20\end{cases}$

J. W. Tanner · Accepted Answer · 2020-01-03T19:44:36.307

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For (c), adding three times the first equation to the third equation yields $0=44$, which is clearly impossible, so there are no solutions.

For (b), subtracting the third equation from the second yields $3x=3$ or $x=1$; thus $3+4y-z=8$ and $5-2y+z=4$; adding those yields $8+2y=12$ or $y=2$ and thus $z=3$, so $(x,y,z)=(1,2,3)$ is the unique solution.

For (a), adding and subtracting the equations we get $2f+g=11$ and $h=f+2$; $f$ could be anything, and then $g$ and $h$ are determined, so there are infinitely many solutions.

edited Jan 03 '20 at 19:44

answered Jan 03 '20 at 18:26

J. W. Tanner

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or $S-T-W=8$ and $S-T-W=\frac{-20}3$ – J. W. Tanner Jan 03 '20 at 18:28
Thank you very much for your clear explanation! Could you also explain how I can identify how many solutions systems a and b have? – Jan 03 '20 at 19:43
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@Vera: I edited my answer to include a and b – J. W. Tanner Jan 03 '20 at 19:45
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Thanks a lot! It's all clear now :) – Jan 03 '20 at 20:09

score 0 · Answer 2 · answered Jan 03 '20 at 18:37

You can use the determinant to evaluate whether the system has unique solutions.

$ det A \neq 0 \iff \text{unique solution}$
$ det A = 0 \implies \text{infinite number of solutions or none}$

The simplest way to determine between infinite or none is to solve and see if the system is inconsistent (results in contradictory constraints).

This answer may be helpful too.

How can you identify how many solutions systems have in $\mathbb{R}^3$?

2 Answers2