Thread: [SOLVED] System of Linear Equations

I have another question regarding another problem. The system is:

x + y = 0
y + z = 0
x + z = 0
ax + by + cz = 0

Find values for a, b, and c so that the system has:
(a) a unique solution
(b) no solution
(c) an infinite number of solutions

I know for a fact it is not possible for this system to have no solution. It has the trivial solution x = y = z = 0. Also, it seems if a + b + c != 0, then it may have an infinite number of solutions. To be honest aside from already knowing part b, I am not quite certain how to proceed. Thanks again for help!

Originally Posted by Alterah

I have another question regarding another problem. The system is:

x + y = 0
y + z = 0
x + z = 0
ax + by + cz = 0

Find values for a, b, and c so that the system has:
(a) a unique solution
(b) no solution
(c) an infinite number of solutions

I know for a fact it is not possible for this system to have no solution. It has the trivial solution x = y = z = 0. Also, it seems if a + b + c != 0, then it may have an infinite number of solutions. To be honest aside from already knowing part b, I am not quite certain how to proceed. Thanks again for help!

Subtract the second from the first:
x + y = 0 Equation1
y + z = 0 Equation2
---------------
x - z = 0 Difference

x + z = 0 Equation3

If x-z =0 AND x+z = 0
THEN there is a finit number of solutions.

ax + by + cz = 0

That makes sense. I apologize if I am a little slow on recognizing a few things. I am very tired at the moment. That makes sense regarding the two equations:

x - z = 0 and x + z = 0. But, to be honest I am not seeing how to link that with a, b, and c.

Of course, x+ z= 0 and x- z= 0 can only be true if x= z= 0. Then x+ y= 0 immediately gives y= 0. That has nothing at all to do with "ax+ by+ cz= 0", except that it is also satisifed by x= y= z= 0. The system has only the single solution x= y= z= 0 no matter what a, b, c are.

Yeah, I realized that shortly before I needed to hand the assignment in. Thanks for the help!