Question:

3 eagerbugs can be on these planes:

z+2y-3x=0, z+2x+y=0, z-2x+8y=0

Can they meet at a point other than origin?

Solution

Looking at the 3 equations we can say that (0,0,0) is definitely a solution of the 3 equations.

We just need to find if they intersect at one more point.

There are more than 1 ways to find it.

Simplest approach would be to solve the 3 equations. So on subtracting second from first we get y=5x and thrid from first we get -6y-x=0, this can happen only if y=0,x=0, so z=0. Hence there is only 1 solution.

Second approach is based on the cramers rule which is used to solve a system of linear equations. We get

|[-3,2,1],[2,1,1],[-2,8,1]|=16!=0

So this equation has only one trivial solution.