Has any method been developed to solve two simultaneous equations in three unknowns? If so, can you explain or supply a link?

tsaeb,

Not really sure of some sophisticated methodology devised,

but surely,
you can easily create three relations,
and successive solutions (in steps)
usually (quite conveniently)
Yields the answer.

Pin~Jinx / anarchist

In general, you're never going to get exactly one solution to a system of equations where you have more variables than equations. If you want to know why, think of it graphically. In the example you mentioned, an equation with 3 variables can be graphically represented by a plane; therefore, you're essentially trying to find where 2 planes intersect. There are 3 ways that 2 planes can intersect: in 0 points (if the planes are parallel), in a plane (if the planes are exactly the same), or in a line, which is probably the most common.

However, with more restrictions on what values the variables can take, you can get more specific answers. Something that comes to mind is if you want to find a solution where all the variables are integers. There are techniques that you can use to find all integer values for x, y, and z such that 56x + 7y - 19z = 8901 These types of systems are commonly referred to as Diophantine Equations. If that's the sort of thing you're wondering about, I can show you a general method to go about solving them.

This is also a common problem in linear algebra, where you can use matrices and row reduction to express the solution of an underdetermined system (more variables than equations) as a linear combination of vectors.

Example: For the following system

x + y + z = 1

x - 3y = 0


We get the matrix

1   1   1   1

1  -3   0   0


     1   1   1   1
=                      (replacing row 2 with row 1 minus row 2)
     0   4   1   1


     1   1   1   1
=                      (dividing row 2 by 4)
     0   1   ¼   ¼


     1   0   ¾   ¾
=                      (replacing row 1 with row 1 minus row 2)
     0   1   ¼   ¼


This gives us the system

x + ¾z = ¾

y + ¼z = ¼


or

x = -¾z + ¾

y = -¼z + ¼


and if we let z = 4t, we get

x = -3t + ¾

y = -t + ¼

z = 4t


or in vector notation,

(x y z)  =  t (-3 -1 4)  +  (¾ ¼ 0),  where t is any real number,

which is just a line in 3-dimensional space.

[This message was edited by silenteuphony on 11-14-02 at 06:32 PM.]

silenteuphony: Gulp! I think that you showed that a plane and a line have in common several points on a line, which points are determinable for successive values of t. Correct me if I am wrong.

Basically, yes. You can actually generate an infinite number of points, since the solution works for any real value of t. Also, the original equations both represent planes, even though one of them only has two variables. Since the whole system is in three variables, the equation x - 3y = 0 represents the plane which extends straight up and down (in the z-direction) from the line in the x-y plane. (I could have used three variables in the second equation, and the same method would still work, but I picked fairly simple equations for my example.)