Question

If
2x+2y=u and
2xy=v
Find
x as a function of u,v
y as a function of u,v

Answer 1

Many ways to do this one.

Answer 2

Square the top equation: \[
(2x+2y)^2=u^2\implies 4x^2+4xy+4y^2=u^2
\]Double the bottom one:\[
2(2xy) = 2v\implies 4xy = 2v
\]Subtract bottom from top: \[
4x^2+4y^2=u^2-2v
\]Divide by \(4\):\[
x^2+y^2=\frac{u^2-2v}{4}
\]This is just a circle.

Answer 3

Hmm, made a small error.

Answer 4

no
x the required is x alone and y alone

Answer 5

Square the top equation: \[
(2x+2y)^2=u^2\implies 4x^2+8xy+4y^2=u^2
\]Quadruple the bottom one:\[
4(2xy) = 4v\implies 8xy = 4v
\]Subtract bottom from top: \[
4x^2+4y^2=u^2-4v
\]Divide by \(4\):\[
x^2+y^2=\frac{u^2-4v}{4}
\]This is just a circle.

Answer 6

How do you know that it is even possible?

Answer 7

i do not know

Answer 8

It's not possible to make \(x\) and \(y\) functions of \(v\) and \(u\).

Answer 9

Because the relationship isn't that of a function.

Answer 10

For something as simple as \(y=x^2\), we can make \(y\) a function of \(x\) but \(x\) is not a function of \(y\) because \(y=1\) has \(x=-1\) and \(x=1\) as solutions.

Answer 11

thank you very much

Answer 12

hmm its just a circl lol
can u solve it -.-