Confusing system of equation with four variables: ab+c+d=3, bc+d+a=5, cd+a+b=2, da+b+c=6

[medali]

it goes as follows:
ab+c+d=3
bc+d+a=5
cd+a+b=2
da+b+c=6

now the main issue I have with this is that I can't get the products to cancel out, and if I try removing one variable such that a=(3-c-d)/b, I'm left with three variables but when I try to reduce the variables again I get a trivial result like 3=3

 

[stapel]

it goes as follows:
ab+c+d=3
bc+d+a=5
cd+a+b=2
da+b+c=6

now the main issue I have with this is that I can't get the products to cancel out, and if I try removing one variable such that a=(3-c-d)/b, I'm left with three variables but when I try to reduce the variables again I get a trivial result like 3=3

To get that sort of trivial result, you're probably re-using an equation at a non-optimal stage of your computations. (I can't see your work, so I can't be sure of this, of course.)

There is a whole-number solution, according to Wolfram Alpha.

What method(s) have you been given for solving this sort of system? Thank you!

 

[medali]

To get that sort of trivial result, you're probably re-using an equation at a non-optimal stage of your computations. (I can't see your work, so I can't be sure of this, of course.)

There is a whole-number solution, according to Wolfram Alpha.

What method(s) have you been given for solving this sort of system? Thank you!

It's an open problem. As for the trivial results, can you explain to me how to avoid it, for instance I replaced a by (3-c-d)/b and the equation when I try to untangle it gives me 3=3.
I've managed to get two 3variable equations so far using transmutation for E1 and vieta's theorem for E2

ab+c+d-bc-d-a=-2
E1: ab+c-b-a=-2

c*d=-a-b+2
c+d=-ab+3
E2: c²+(ab-3)c-ab+2

Normally I would need a third equation with the variables a,b,c but I don't really know if what I've done was even necessary. First time encountering this kind of problem and again, eveything I try other than that ends up in a loopwhole.

 

[khansaheb]

It's an open problem. As for the trivial results, can you explain to me how to avoid it, for instance I replaced a by (3-c-d)/b and the equation when I try to untangle it gives me 3=3.
I've managed to get two 3variable equations so far using transmutation for E1 and vieta's theorem for E2

ab+c+d-bc-d-a=-2
E1: ab+c-b-a=-2

c*d=-a-b+2
c+d=-ab+3
E2: c²+(ab-3)c-ab+2

Normally I would need a third equation with the variables a,b,c but I don't really know if what I've done was even necessary. First time encountering this kind of problem and again, eveything I try other than that ends up in a loopwhole.

You want us to "catch" your mistake (if there is any)!

To do that - we need to 'see' your work. A well-lit legible picture of your (handi)work will be sufficient.

 

[khansaheb]

it goes as follows:
ab+c+d=3
bc+d+a=5
cd+a+b=2
da+b+c=6

now the main issue I have with this is that I can't get the products to cancel out, and if I try removing one variable such that a=(3-c-d)/b, I'm left with three variables but when I try to reduce the variables again I get a trivial result like.

These are NOT LINEAR equations - This set may not have unique solution.

 

[medali]

These are NOT LINEAR equations - This set may not have unique solution.

Do you mean they have infinite solutions? Because I'm 100% sure this problem is solvable. I already wrote out my work but I'm short one equation

I've managed to get two 3variable equations so far using transmutation for E1 and vieta's theorem for E2

ab+c+d-bc-d-a=-2
E1: ab+c-bc-a=-2

c*d=-a-b+2
c+d=-ab+3
E2: c²+(ab-3)c-ab+2