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I have tried this and got $a=-15$ and $a=-9$; is that right?

Consider the system of linear equations given by:

$$\begin{align} −2x+5y+4z&=1\\ −4x+(12+a)y+10z&=1\\ (5+a)x−3y−8z&=1 \end{align}$$

(a) Determine all values of $a$ for which the system has a unique solution.
(b) Determine the value of $a$ for which the system is inconsistent.
(c) Determine the value of $a$ for which the system has infinitely many solutions.

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  • $\begingroup$ is that okay now ? $\endgroup$ – ellen Dec 12 '17 at 14:10
  • $\begingroup$ So your solution $a=-15$ is a solution for which question?! $\endgroup$ – 5xum Dec 12 '17 at 14:12
  • $\begingroup$ Also, how did you come to the solution? $\endgroup$ – 5xum Dec 12 '17 at 14:14
  • $\begingroup$ yes and when i sub those in for x,y,z i get 0=0 . why do i have to use the det? $\endgroup$ – ellen Dec 12 '17 at 14:33
  • $\begingroup$ a = 5/2 and a= -3 ?? $\endgroup$ – ellen Dec 12 '17 at 14:43
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HINT

Let's consider the Augmented Matrix for the system

$$\left[\begin{array}{ccc|c} -2& 5& 4& 1\\ -4& 12+a& 10& 1\\ 5+a& -3& -8& 1 \end{array}\right]$$

And then proceed by row operation to obtain the Row Reduced Echelon Form.

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@ellen Use Cramer's rule here

You can also solve it by Martin's rule via Matrix Method.

If you follow those steps in the methods I mentioned, you should get a correct answer for sure

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