I would be very grateful if someone could help me with this system of equations.

[alejandroid]

Hello everyone, I would like someone to help me to know if my answer to this system of equations is correct, and if not, what is the correct answer:


The exercise says determine the values of k so that the system has:

a) infinitely solutions
b) has no solution
c) has only one solution

And this was my answer:

 

[Cubist]

You seem to have the correct answer, but there seems to be something missing from your explanation. Why did you write...

[math]k^2-k-156=0[/math]
Maybe you were thinking that if [imath]c_1z=c_2[/imath], then in order to obtain infinite values of z this implies [imath]c_1=0[/imath] and [imath]c_2=0[/imath]? But how did you obtain k=13 for the "no solution" scenario?

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I would have approached this by factoring the WHOLE of the following...

[math]\left( k^2-k-156 \right)z=k^2-144[/math]
Using your observations...

[math]\left((k-13)z\right)(k+12)=(k-12)(k+12)[/math]
[math](?)(k+12)=0[/math]
...continue