Number of solutions for system of equations

[bargaj]

Hello!

I have a simple question about solutions, better said number of solutions for this system of equations.
[math]\begin{cases} x_{1 } − x_{2 } + 3x_{3 } − 2x_{4 } = 1\\ −2x_{1 } + 2cx_{2 } − 4x_{3 } + 2x_{4 } = −7\\ − 2x_{3 } + (−c + 6)x_{4 } = 2c + 15\\ − 2x_{3 } + c^{2 }x_{4 } = c^{2 }\end{cases}[/math]
I know it's only possible that this system has either 0, 1 or [imath]\infty[/imath] number of solutions, for different values of c:
[math]c = -3 \rightarrow \infty\\ c = 1 \rightarrow \infty\\ c = 2 \rightarrow 0 \\ c \in ℝ \setminus \{-3, 1, 2\} \rightarrow 1[/math]
My question is: for which c has this system at the utmost 2 solutions? Should it be only for when the whole system has only one solution or also when it has none? Thank you for your help!

 

[blamocur]

I must be missing something: since you know that the system can only have 0,1 or [imath]\infty[/imath] solutions why do expect to find [imath]c[/imath] which leads to 2 solutions?

 

[bargaj]

I'm sorry if the question wasn't clear enough. English is my second language so I struggle a bit to translate the assignment from my mother language. To expand on it: I'm trying to find for which values of c will the system have up to 2 solutions. Not exactly 2. I'm confused if this means that those values don't exist, because the system cannot have 2 solutions, or if this means that it can have 0 or 1 solution, meaning that the answer would be c∈R ∖ {−3, 1}. Really thankful for your feedback!

 

[Dr.Peterson]

I'm sorry if the question wasn't clear enough. English is my second language so I struggle a bit to translate the assignment from my mother language. To expand on it: I'm trying to find for which values of c will the system have up to 2 solutions. Not exactly 2. I'm confused if this means that those values don't exist, because the system cannot have 2 solutions, or if this means that it can have 0 or 1 solution, meaning that the answer would be c∈R ∖ {−3, 1}. Really thankful for your feedback!

It may be helpful to show us the original language of the problem so we can try to determine what it actually means; but if it asks for the condition under which there will be "at most 2 solutions", or "no more than 2 solutions" (or "less than 2 solutions"), I would expect that to mean "0 solutions or 1 solution", that is, not infinitely many.

I'm not sure what you mean by "those values don't exist".

 

[Steven G]

Maybe this will help. If there are two or more solutions then there are infinitely many solutions.