An infinite number of solutions

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Quandaries & Queries

Question from Sean, a student:

this is a linear equations problem; my teacher gave me the solution worked out but I
am not sure how she solved it exactly

first:
3535.5 + Fbd (.866) + Fbc (.5) - Fab (.5) = 0
and
-3535.5 - Fab (.866) - Fbc (.5) - Fbd (.5) = 0

next:
3535.5 + Fbd (.866) - Fab (.5) = 0
3535.5 - Fbd (.5) - Fab (.866) = 0
-6123.5 - Fbd (1.5) + Fab (.866) = 0

-9659 = Fbd 2.0

and the answers are:

Fbd = -4829.5
Fab = - 1293.7

could you please tell me how to solve this in the traditional linear systems method

Sean,

You are given the equations

3535.5 + Fbd (.866) + Fbc (.5) - Fab (.5) = 0
-3535.5 - Fab (.866) - Fbc (.5) - Fbd (.5) = 0

Since you have more unknowns than equations you are not going t obtain a unique solution.

My first action was to rearrange the two equations to obtain

3535.5 - Fab(.5) + Fbc (.5) + Fbd (.866) = 0
-3535.5 - Fab (.866) - Fbc (.5) - Fbd (.5) = 0

and then add the equations to get

-Fab(1.366) + Fbd(.366) = 0

Fbd = Fab(0.3732)

If you substitute this back into the two given equations they both simplify to the same equation

3535.5 + Fab(2.732) + Fbc(0.5) = 0

Fbc = -7071 - Fab(5.464)

Hence you can give Fab any value and obtain a solution with that value for Fab and Fbd and Fbc determined by the highlighted equations.

Harley

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