Solve for k in System of Equations

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Discussion Overview

The discussion revolves around finding the value of k in a system of equations represented in matrix form. Participants explore the implications of the determinant of the matrix on the existence of solutions, considering both unique and infinite solutions.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that the determinant of the matrix plays a crucial role in determining the conditions under which the system has a solution.
  • One participant suggests that the matrix has a unique solution if the determinant is nonzero, while if the determinant is zero, there may be no solutions or infinitely many solutions.
  • There are two approaches mentioned: one is to find k that makes the determinant nonzero for a unique solution, and the other is to find k such that the determinant is zero for infinitely many solutions.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the specific value of k or the implications of the determinant, indicating multiple competing views and unresolved aspects of the discussion.

Contextual Notes

The discussion does not provide specific values or conditions for k, and there are unresolved mathematical steps regarding the calculation of the determinant.

wonguyen1995
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Find k to have a solution?
x-3y=6
x+3z=-3
2x+kx+(3-k)z=1LATEX
 
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wonguyen1995 said:
Find k to have a solution?
x-3y=6
x+3z=-3
2x+kx+(3-k)z=1LATEX

Have you thought about what role the determinant would play?
 
dwsmith said:
Have you thought about what role the determinant would play?

of course I think it should better if i have sample of solution. i will research carefully on this.
 
wonguyen1995 said:
of course I think it should better if i have sample of solution. i will research carefully on this.

We know the matrix
\[
\begin{bmatrix}
1&-3&0\\
1&0&3\\
2&k&3-k
\end{bmatrix}
\]
has unique solution if the determinant is what?
Second, if the determinant is zero, we have no solutions or infinitely many solutions.

We have two approaches. One assume the determinant is nonzero and find k that makes it invertible or assume the determinant is zero and try to find a k such that we have infinitely many solutions.
 

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