SOLUTION: Can 8x-2y+3z-7x+6y-7w=39 be solved or just simplified?

Question 346561: Can 8x-2y+3z-7x+6y-7w=39 be solved or just simplified?
Answer by ptaylor(2198) (Show Source): You can put this solution on YOUR website!
Just simplified.
8x-2y+3z-7x+6y-7w=39 first, lets collect like terms:
8x-7x-2y+6y+3z-7w=39 or
x+4y+3z-7w=39 Now we have 1 equation with 4 unknowns (and no restrictions on the numbers) and there will be an infinite number of solutions. In most all cases, when you have more unknowns than equations, there is a lot of trial and error involved in finding the solutions, if a finite number of solutions exist. There is almost always more than one solution. Lets, take a simple example:
Solve:
x+y=6
Here are some solutions:
x=0, then y=6
x=1, then y=5
x=2, then y=3
x=3 then y=3
x=4, then y=2
x=5, then y=1
x=6, the y=0
These are just whole, positive number solutions. If we start looking at fractions and negative numbers, we clearly have an infinite number of solutions.
There are lots of problems floating around that result in more unknowns than equations but the answers will usually be constrained to whole, positive numbers. If we constrain the above problem (x+y=6) to whole, positive numbers, then we only have six solutions.
Hope this helps---ptaylor
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