Given the equation: ( 2 , − 14 , 8 ) x + ( − 3 , 21 , − 12 ) y = 0 → Find the scalars x and y, not both zero, such that the above equation holds.My attempt:I tried solving the equation system 2x−3y=0,−14x+21y=0,8x−12y=0 but I only obtained trivial solutions, i.e. x=0,y=0

Question

Given the equation:  (  2  ,  −  14  ,  8  )  x  +  (  −  3  ,  21  ,  −  12  )  y  =            0      →      Find the scalars x and y, not both zero, such that the above equation holds.My attempt:I tried solving the equation system 2x−3y=0,−14x+21y=0,8x−12y=0 but I only obtained trivial solutions, i.e. x=0,y=0

Bradley Sherman · Accepted Answer

Your effort makes a lot of sense! But you not only find the trivial solution: your system of equations is correct and it is equivalent to the only equation:  2  x  −  3  y  =  0  ,  whose solution is  y  =      2    3    x  .  So, all the ordered couples of the kind   (  x  ,  y  )  =  (  x  ,      2    3    x  ) are solutions.The trivial solution (0,0)  is an example, other ones are   (  1  ,      2    3    ) and (3,2)  or (−3,−2) but one linear equation in two variables has an infinity of such ordered couples of values as solutions.