How to isolate one of the terms or derive an expression to arrive at one of them from given expressions

Question

So I have been given 4 expressions and in three variable $a,b$ & $c$. Using these three expressions I have to isolate either of $a,b$ or $c$. However, I have not been able to come up with any equation or expression using which I could possibly isolate one of the terms. By isolate, I mean that given any expression in terms of some variables I should be able to do some operations by which I am able to get one or two variables. For example:

If $\;\text{eq1} = a+b+c \\ \;\text{eq2} = a+b$

Then I must be able to obtain $a,b$ or $c$ by doing some operations and in this example it is simple subtraction.

Expressions given:

$\text{Ex1} = a + b + 2c \\ \text{Ex2} = c^2 + ab + 2bc + 2ca \\ \text{Ex3} = 2abc + ac^2 + bc^2 \\ \text{Ex4} = 4((c−a)^2(c−b)^2(4c^2−4bc−4ac+9b^2−14ab+9a^2))$

Expressions I arrived at:

$\text{Ex5} = a^2 + b^2 + 2c^2 \\ \text{Ex6} = 3a^2 + 3b^2 + 4c^2 - 2ab - 4ac - 4bc$

I have not been able to arrive at any solution till now. I think the answer might be to use the expressions to derive some expression for Ex4 and divide it with that to arrive at the answer.

Edit: Wrong terminology was used. Corrected it. Edit2: The goal is to manipulate the various equations so that after some operations are done on them the result is a,b or c while we only have the above expression to get our answer with.

Answer 1

$$ \begin{align} u &= a + b + 2c \\ v &= c^2 + ab + 2bc + 2ca \\ w &= 2abc + ac^2 + bc^2 \\ z &= 4((c−a)^2(c−b)^2(4c^2−4bc−4ac+9b^2−14ab+9a^2)) \end{align} $$

The question is not clear and can be interpreted in different ways.

• Calculate $\,z\,$ given the values of $\,u,v,w\,$. $\;-\;$ This can be done by eliminating $\,a,b,c\,$ between the four equations, and results (courtesy WA) in: $$ z = -4 (27 u^3 w - 9 u^2 v^2 - 108 u v w + 32 v^3 + 108 w^2) \tag{*} $$

• Solve the system for $\,a,b,c\,$. $\;-\;$ A system with $4$ equations in $3$ unknowns is overdetermined in general. The condition for solutions to exist is that $(*)$ be satisfied, and in that case eliminating $\,b,c,\,$ between the first $3$ equations gives a sextic equation in $a$ alone (WA): $$ 16 a^6 - 24 a^5 u + a^4 (3 u^2 + 32 v) + a^3 (2 u^3 - 12 u v - 40 w) \\+ a^2 (3 u^4 - 18 u^2 v + 30 u w + 20 v^2) + a (-2 u^3 v + 6 u^2 w + 8 u v^2 - 36 v w) \\+ 4 u^3 w - u^2 v^2 - 18 u v w + 4 v^3 + 27 w^2 = 0 $$