The following system of linear equations is said to be underdetermined because t

Question

The following system of linear equations is said to be underdetermined because there are more
unknowns than equations.

x_1 +2x_2 - 3x_3 = 4
2x_1 - x_2 + 4x_3 = -3

(the x_1, x_2, etc. are variables x1 and x2. Not sure how to get the numbers as subscripts)

Similarly, the following system is overdetermined because there are more equations than unknowns.

x_1 +3x_2 = 5
2x_1 - 2x_2 = -3
-x_1 +7x_2 = 0

You can explore whether the number of unknowns and the number of equations have any bearing on the
consistency of a system of linear equations.

1. Can you find a consistent underdetermined linear system? If so, give an example and solve it.
Otherwise, explain why it is impossible.
2. Can you find a consistent overdetermined linear system? If so, give an example and solve it.
Otherwise, explain why it is impossible.
3. Can you find an inconsistent underdetermined linear system? If so, give an example, explaining
why it has no solution. Otherwise, explain why it is impossible.
4. Can you find an inconsistent overdetermined linear system? If so, give an example, explaining
why it has no solution. Otherwise, explain why it is impossible.
5. Explain why you would expect an overdetermined linear system to be inconsistent. Must this
always be the case?
6. Explain why you would expect an underdetermined linear system to have an infinite number of
solutions. Must this always be the case?

Explanation / Answer

Can you find a consistent underdetermined linear system? Consistent = 1+, underdetermined = infinite. So sure - your first example. x+y+z=0 x+y=20 2. Can you find a consistent overdetermined linear system? If so, give an example and solve it. sure. Just two+ of them need to be the same. x+y=2 2x+2y=4 3x+3y=6 3. Sure by your definition - x+y+z+d=7, 2x+y+z+d=7, x=1 4. Sure. x+y+z+d=7, 2x+y+z+d=7, x=1, y=2, z=3 5. Because there are usually to many constraints on the variables. No, #2. 6. Because there aren't enough constraints on the variables. And no. Look at #3.

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