Finding a system of linear equations from solutionsIs there a simple way to find the system of linear equations given the solutions? For example, find a system with 2 equations and 3 variables that has solutions (1, 4, -1) and (2, 5, 2). { x 1 − x 2 = − 3 3 x 1 − x 3 = 4 The method I used to get this answer was guessing and checking, but surely there is a simpler and more efficient way to go about it?

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Finding a system of linear equations from solutionsIs there a simple way to find the system of linear equations given the solutions? For example, find a system with 2 equations and 3 variables that has solutions (1, 4, -1) and (2, 5, 2).      {                                        x            1                    −                      x            2                    =          −          3                                      3                      x            1                    −                      x            3                    =          4                        The method I used to get this answer was guessing and checking, but surely there is a simpler and more efficient way to go about it?

Judovh0 · Accepted Answer

The solution set of your system of equations is not simply   {  (  1  ,  4  ,  −  1  )  ,  (  2  ,  5  ,  2  )  }.For example,   (  0  ,  3  ,  −  4  ) is a solution. There are infinitely many others.And we cannot obtain a system of two linear equations in three unknowns whose solution set is   {  (  1  ,  4  ,  −  1  )  ,  (  2  ,  5  ,  2  )  }. For if the system has a solution, it has infinitely many.As to your question about producing such a system in a systematic way, let the two equations be       a    1    x  +      b    1    y  +      c    1    z  =      d    1  ,       a    2    x  +      b    2    y  +      c    2    z  =      d    2  . The conditions are      a    1    +  4      b    1    −      c    1    =      d    1  ,   2      a    1    +  5      b    1    +  2      c    1    =      d    1  ,      a    2    +  4      b    2    −      c    2    =      d    2  ,  2      a    2    +  5      b    2    +  2      c    2    =      d    2  .We have   4 equations in   8 unknowns, so obviously a lot of slack. One way to cut down on it is to decide that the       d    i   will be   0.Choose say       a    1    =  0, and solve for       b    1   and       c    1  ; Choose       b    2  , and solve for       a    2   and       c    2  .

Finding a system of linear equations from solutions Is there a simple way to find the system of lin

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