Linear problems

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May 2nd 2006, 02:30 PM
LilDragonfly

Linear problems

Please help me solve these, thank you so much!

1.
Solve for (x,y,z)
2x + 5y + 3z = 16
x/5 + y/2 + z/3 = 2
3x - 2y - 4z = -2

State clearly whether this system is consistent or inconsistent. State whether the solution is unique, or if there are an infinite number of solutions, or no solution at all. If the solution is unique, state the solution.

2.
The equations of three planes are:
3x + 2y + 7z = 25 (1)
2x + y + 4z = 14 (2)
5x + 3y + 11z = 39 (3)

Solve the above system of equations. State whether this system is consistent or inconsistent and how many solutions there are.
May 2nd 2006, 02:55 PM
ThePerfectHacker

Quote:

Originally Posted by LilDragonfly

Please help me solve these, thank you so much!

1.
Solve for (x,y,z)
2x + 5y + 3z = 16
x/5 + y/2 + z/3 = 2
3x - 2y - 4z = -2

State clearly whether this system is consistent or inconsistent. State whether the solution is unique, or if there are an infinite number of solutions, or no solution at all. If the solution is unique, state the solution.

I do not know how you solve this, but I look at the determinant. Which is approximately, .633 since it is non-zero the solution is unique.
It solution is,
Using my matrix calculator I get,
$(x,y,z)=(10,-8,12)$
May 2nd 2006, 06:09 PM
LilDragonfly

Thank you ThePerfectHacker for that, but is there anyone else that has a greater understanding on this topic to provide a fuller explanation on how to answer these two questions?
May 2nd 2006, 06:39 PM
ThePerfectHacker

Quote:

Originally Posted by LilDragonfly

Thank you ThePerfectHacker for that, but is there anyone else that has a greater understanding on this topic to provide a fuller explanation on how to answer these two questions?

Should have not I explained it with determinants?

Maybe you wanted me to find the inverse of the augmented matrix?
May 4th 2006, 07:55 PM
LilDragonfly

I don't think I really explained the question properly, so I have found a few statements on the solutions involved: unique, inconsistent..etc. To be quite honest, I have no idea what it all means but I hope it is some help for you. I would love to understand how you answer question one and two, so if you can explain it in lay-mans terms then I will be a very happy and rather informed girl :)

- When there is a solution or many solutions to a system of simultaneous equations they are said to be consistent equations. When there is no solution the equations are said to be inconsistent.

When solving simultaneous equations with two unknowns:
- one solutions means the lines intersect
- a true statement (eg 0 = 0) means there is an infinite number of solutions, which means the lines are coincident.
- a false statement means there are no solutions, this means the lines are parallel.

When solving simultaneous equations with three unknowns:
- a false statement from any pair of the three equations means there are no solutions, there are various possibilities for the relative positions of the planes.
- no false statements and no unique solution means there are an infinite number of solutions – here 3 planes intersect on a line or 2 planes are coincident with one intersecting them or 3 coincident planes.

If k = n and the matrix A is non-singular, then the system has a unique solution in the n variables. In particular, there is a unique solution if A has a matrix inverse A^-1. In this case, x = A^-1 b. (http://mathworld.wolfram.com/LinearS...Equations.html)

Thank you!