Constraint on a linear system?

[bonfire09]

The question goes like this.
Among all solutions that satisfy the homogeneous system
x + 2y + z = 0
2x + 4y + z = 0
x + 2y − z = 0
determine those that also satisfy the nonlinear constraint y − xy = 2z

I know that one of the solutions [0,0,0] but I'm not sure how to find the others. I row reduced the linear system and its general solution is x=y[-2,1,0].

 

[Mark44]

The question goes like this.
Among all solutions that satisfy the homogeneous system
x + 2y + z = 0
2x + 4y + z = 0
x + 2y − z = 0
determine those that also satisfy the nonlinear constraint y − xy = 2z

I know that one of the solutions [0,0,0] but I'm not sure how to find the others. I row reduced the linear system and its general solution is x=y[-2,1,0].

So the general solution to your system of equations is t<-2, 1, 0>, where t is any real number. Another way to write this is <-2t, t, 0>. Does this satisfy your constraint for some value of t?

 

[bonfire09]

oh -1/2<-2,1,0>=<1,-1/2,0> is another solution to the constraint.

 

[Mark44]

Are you sure?

 

[bonfire09]

i think so.

 

[HallsofIvy]

You have already said that any solution to the linear equations must have x= -2t, y= t, z= 0. (I did not check that.)

Now you also want to require that y- xy= 2z which is just t- (2t)(t)= 0 or [itex]t- 2t^2= 0[/tex]. That's a quadratic equation and has two solutions.