MHB Determining when a system of equations has no solution and infinite solutions

[Ankit2]

Find the surface in terms of a,b,c on which the following system of equations has no
solution
ax-2y+3z=5
-x+y-bz=-3
2x+cy-2z=d
Could there be any values of a,b,c,d for which the system has infinite solution? (Justify).

 

[MarkFL]

Hello and welcome to MHB! :D

I have retitled your thread so that it indicates the nature of the question being asked.

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

 

[HallsofIvy]

Find the surface in terms of a,b,c on which the following system of equations has no
solution
ax-2y+3z=5
-x+y-bz=-3
2x+cy-2z=d
Could there be any values of a,b,c,d for which the system has infinite solution? (Justify).

Have you tried to solve the system? The first thing I notice is that if you multiply the second equation by "a" and add that to the first equation, you eliminate "x": (ax- 2y+ 3z)+ (-ax+ ay- abz)= (2- a)y+ (3- ab)z= 5- 3a. And that if you multiply the second equation by "2" and add that to the third equation, you also eliminate x: (2x- cy- 2z)+ (-2x+ 2y- 2bz)= (2- c)y- (2+ 2b)z= d- 6. Can you solve those two equations for y and z? If not what would stop you?