The following two vectors satisfy the same system of linear equations. u=⎡⎣⎢⎢6-5-5⎤⎦⎥⎥, v=⎡⎣⎢⎢746⎤⎦⎥⎥ Find x and y that make the vector ⎡⎣⎢⎢3xy⎤⎦⎥⎥ a solution to the corresponding homogeneous system:

x=-30/71, y=0

Step-by-step explanation:

if a vector satisfies an equation of the form ax+by+cz=0 then the vector is

parallel to the plane ax+by+cz=0, and, the cross product of two vectors results in an orthogonal vector to both.

So, the normal vector of the plane ax+by+cz=0, can be found as the cross product of the two parallel vetors to the plane:

= * =

so, the homogeneous system is:

-10x-71y+59z=0

by replacing the vector in the system

-30-71x+59y=0

So, there are 2 unknown variables and 1 equation, it means that 1 variable is free

so, y=0 is a random defition and x can be obtain with the equation

-30-71x=0 - > x=-30/71