Under what condition on the constans a,b,and c does the system 2x-y+3z=a, x+2y+z=b, and 7x+4y+9z=c has a unique solution? No solution? Infinitely many solutions?

Question

zehanz · Answer

Are you familiar with the fact that these equations describe (flat) planes in three dimensional space?

zehanz · Answer

The points in such a plane have x, y and z coordinates that satisfy the equation.
A solution of the three equations as a system must be points that lie on all three planes!

Can you imagine a situation when there is a unique solution, i.e. there is exactly one point lying on all of the three planes?

anonymous · Answer

I know that. My question is using matrix how do I define the values of a,b,and c that makes the system (the three equations) to have either one solution, no solution, or infinitely many solutions.

zehanz · Answer

Oh, I'm sorry - not familiar enough with matrices to use them with this prob. :(

zehanz · Answer

But - if I look at the equations, I see three planes, neither of which is parallel with another. They all have a different orientation (or: their normal vectors are independent).
This means there is always a unique solution, because changing a, b or c only displaces a plane without changing its orientation.

So: any value of a, b and c gives a unique solution (x, y, z). The planes have always exactly one point in common.