Will the bugs meet?

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.

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