Algebra - Linear systems to solve problems

[Kiwigirl]

Please help me with these, I have partly answered question 1 on a Matrix calculator, so (x,y,z) = (10, -8, 12).

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.

 

[tkhunny]

Those are lovely problem statements.

What are the criteria for "consistent", "Unique", etc?

 

[Kiwigirl]

Hi there tkhunny, I found some info for your question, although to be quite honest, I have no idea what it all means but I hope it is some help to you.

- 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/LinearSyst ... tions.html)

 

[tkhunny]

Let's glean a little bit.

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.

One solutions means Unique

A true statement (eg 0 = 0) means there is an infinite number of solutions

A false statement means there are no solutions, this means the lines are parallel.

OK. Now solve your systems and see which results you get.

Is there a solution? (Consistent or Inconsistent)
If so, how many? (Unique or Not Unique)