I want to show that the following system of equations does not have a solution, but I do not know how to do this w 1 + w 2 = 1 2 w 1 s 1 + w 2 s 2 = 1 6 w 1 t 1 + w 2 t 2 = 1 6 w 1 s 1 t 1 + w 2 s 2 t 2 = 1 24 w 1 s 1 2 + w 2 s 2 2 = 1 12 w 1 t 1 2 + w 2 t 2 2 = 1 12 that all t 1 , t 2 , w 1 , w 2 , s 1 , s 2 are unknown.

Question

I want to show that the following system of equations does not have a solution, but I do not know how to do this      w    1    +      w    2    =      1    2        w    1        s    1    +      w    2        s    2    =      1    6        w    1        t    1    +      w    2        t    2    =      1    6        w    1        s    1        t    1    +      w    2        s    2        t    2    =      1    24        w    1        s    1    2    +      w    2        s    2    2    =      1    12        w    1        t    1    2    +      w    2        t    2    2    =      1    12  that all       t    1    ,      t    2    ,      w    1    ,      w    2    ,      s    1    ,      s    2   are unknown.

Scarlet Reid · Accepted Answer

First note that we cannot have       w    1    =  0 or       w    2    =  0.Subtracting the second and third give us      w    1    (      s    1    −      t    1    )  =  −      w    2    (      s    2    −      t    2    )Now if       s    1    =      t    1   then       s    2    =      t    2  , and we can verify there is no solution, by comparing the fourth and fifth.Subtracting the fifth and sixth gives us      w    1    (      s    1    2    −      t    1    2    )  =  −      w    2    (      s    2    2    −      t    2    2    )and thus from the above, we must have that      s    1    +      t    1    =      s    2    +      t    2    =  x    (  say  )Adding second and third gives us  (      w    1    +      w    2    )  x  =      1    3  and so  x  =      2    3  using the first equation.Adding the fifth, sixth and two times the fourth gives us      w    1        x    2    +      w    2        x    2    =      1    4  and so  x  =            ±      1              2      a contradiction.

Anahi Jensen · Accepted Answer

Good answer

I want to show that the following system of equations does not have a solution, but I do not know ho

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New Questions in Algebra I