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Homework 1

# Homework 1 - and not for the other 3 Solve the following...

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1. Solve the system of equations in the scrap metal blending problem using Matlab, Math- ematica, Mapel, or Excel, ignoring the nonnegativity restrictions on the variables. Is the solution to the system of equations unique? See if you get a solution satisfying the nonneg- ativity restrictions anyway. 2. Need a nonnegative solution to the following system of linear equations. x 1 - 2 x 2 + x 3 + x 4 - x 6 + 6 x 7 = 4 x 1 - x 2 - 2 x 3 - x 4 + x 5 + 8 x 6 - 17 x 7 = - 1 2 x 2 + x 3 - 2 x 4 - 5 x 5 + 3 x 6 - 4 x 7 = - 3 Solve using any linear equation software package, ignoring nonnegativity restrictions on variables. Does this system of equations have alternate solutions? If so, what is the dimension of the set of all solutions to this system of equations? Explain. Did you get a solution satisfying the nonnegativity restrictions? Explain the diFerence in the performance of these software packages on problems 1 and 2. Basically, you have to explain why you were able to get a nonnegative solution for one problem using this software,
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Unformatted text preview: and not for the other. 3. Solve the following systems of linear equations by hand using the GJ elimination method. Use pivot elements of 1 or-1 as far as possible to avoid the occurrence of fractions. If system inconsistent, obtain a solution to the alternate system in each case. How many solutions of the alternate system were you able to obtain? ( a )-x 2 + 3 x 3 + 8 x 4-7 x 6 =-3 7 x 1 + x 2-2 x 3-10 x 4 + 2 x 5-3 x 6 =-4 10 x 1 + 2 x 3 + 12 x 4 + 5 x 5-8 x 6 = 2 ( b )-5 x 1-2 x 2 + 10 x 3 + x 4 + 7 x 5 + 2 x 6-4 x 7 = 2-3 x 1 + 3 x 2-2 x 3-x 4 + 2 x 5 + x 6 + 3 x 7 = 3-8 x 1 + x 2 + 8 x 3 + 9 x 5 + 3 x 6-x 7 = 6 x 1 + 2 x 3-3 x 5-4 x 6 + 2 x 7 = 5-7 x 1 + x 2 + 10 x 3 + 6 x 5-x 6 + x 7 = 8 1...
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