x + 2y + z = 0 and y + 2z = 0 ?

Question

I do not even know how this one works

Answer 1

x + 2y + z = 0

      y + 2z = 0

 

Since y + 2z = 0, y = -2z

 

Substituting for y in the first equation, we have x + 2(-2z) + z = 0

 

                                                                           So, x = 3z

 

x = 3z

y = -2z

z = any real number

 

Solution set:  {(3z, -2z, z) l z = any real number}

 

There are infinitely many solutions.  For example, if z = 1, we get (3, -2, 1).  If z = 0, we get (0,0,0), etc.  

 

Geometrically, the graphs of x + 2y +z = 0 and y + 2z = 0 are planes in 3-space.  The solution set of the given system is the line of intersection of the two planes.