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I'm trying to solve this problem, which basically states this:

Based on the system of equations of the variables $x, y$ and $z$: $$\begin{cases} x y - 2\sqrt{y} + 3 y z = 8 \\ 2 x y - 3\sqrt{y} + 2 y z = 7 \\ - x y + \sqrt{y} + 2 y z = 4 \\ \end{cases}$$please tell which of the next alternatives are correct:

  • The system is possible and determined
  • The range of the extended matrix of the system is 2.
  • On the transposed matrix of the coefficients, associated with the system: $a_{12} = -3$
  • The coefficient matrix associated to the system is non invertible.

So, what I did was construct the extended matrix from the equation system given in the problem statement, replacing the variables as this:

  • $a = xy$
  • $b = \sqrt{y}$
  • $c = yz$

The matrix constructed is this: \begin{bmatrix} 1 & -2 & 3 & 8 \\ 2 & -3 & 2 & 7 \\ -1 & 1 & 2 & 4 \end{bmatrix} Solving by gaussian method, the matrix is reduced to: \begin{bmatrix} 1 & -2 & 3 & 8 \\ 0 & 1 & -4 & -9 \\ 0 & 0 & 1 & 3 \end{bmatrix}

So finally:

  • $a = 5$
  • $b = 3$
  • $c = 3$

The values of $x, y$ and $z$ could be calculated using the previous equations, giving this values as results:

  • $x = 5/9$
  • $y = 9$
  • $z = 1/3$

This is the far as I can go on solving this problem, my question basically is, how to transform this to a linear equation system of x, y and z, because I do have the values of $x, y$ and $z$ but the statements that I have to indicate if they are correct or not, are based on that matrix that I don't know how to build.

I was thinking that maybe this problem doesn't look for me to build such a matrix, instead, given that there exist a solution for the system proposed initially, I should deduct the veracity of the statements:

  • First statement is correct as each variable has just one unique value.
  • Second statement is incorrect because for each variable having one single value, the range of the matrix should be 3.

Second and third statement I have no idea how to answer to.

Please give me a hand on this. Thank you a lot.

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  • $\begingroup$ It is important to include the entire question as text, instead of as images. I added the necessary MathJax for you. $\endgroup$ – Nominal Animal Jan 13 at 0:56
  • $\begingroup$ Thank you for including it! $\endgroup$ – Carlos Córdova S. Jan 13 at 0:58
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    $\begingroup$ I suspect that the matrix-related items refer to the coefficients of the linear "$a$-$b$-$c$" system you created by assigning $a=xy$, $b=\sqrt{y}$, $c=yz$. $\endgroup$ – Blue Jan 13 at 1:13
  • $\begingroup$ It is not, the problem said "Based on the equation system of the variables "x-y-z" $\endgroup$ – Carlos Córdova S. Jan 13 at 1:15
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    $\begingroup$ @CarlosCórdovaS.: Well, the $x$-$y$-$z$ system is hopelessly non-linear, so there's no appropriate interpretation of "matrix of coefficients" for it. (The work you did in assigning $a$, $b$, $c$ seems exactly the correct approach.) $\endgroup$ – Blue Jan 13 at 1:20

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