finding constants for any out of five variables help

[Guest]

I have five equations:

yz = a(2z + y) + 2u.......[1]

xz = a(2z + x) + 2u.......[2]

xy = a(2x + 2y) + 8u.....[3]

2xz + 2yz + xy = 1500...[4]

2x + 2y + 8z = 200........[5]

and from [5] I've determined that

x = 100 - y - 4z

y = 100 - x - 4z

z = 25 - (1/4)x - (1/4)y

But after plugging them into multiple different equations I haven't come up with a value for anything. If anybody can help me along with this (just set me in a path that would work for finding just one of the variables if you could) I'd be greatful.

 

[tkhunny]

1) Non-linear simultaneous equations can be trouble.
2) Playing with one equation only will not produce much fruit.
3) There are various ways to go about it, but if one is courageous, and can wade through a deluge of algebra, sometimes one can find an actual solution - sometimes even unique!

One thing you can exploit in this set of equations is that it is symmetric in 'x' and 'y'. Change every 'x' to 'y' and every 'y' to 'x' and you get the same thing! That will help.

You can also simplify the last one. x + y + 4x = 100 is sufficient.

I get \(\displaystyle x = y = 50 + 10\sqrt{10}\). Let's see what you get for the rest.

 

[Denis]

yz = a(2z + y) + 2u.......[1]
xz = a(2z + x) + 2u.......[2]

Subtract to get a(2z + x) - a(2z + y) = xz - yz
Simplify that and you'll get a(x - y) = z(x - y), so a = z

As TK told you, "play around"....