Finding inverse of non-linear transformation

[snoggerT]

Find the inverse of the (nonlinear) transformation from R^2 to R^2 given by

u=3y
v=3x^7-6y

x=?
y=?

The Attempt at a Solution

- I'm really not sure what to do on this problem. We haven't seen any problems even similar to it in class, so I'm looking for help on it.

 

[HallsofIvy]

Find the inverse of the (nonlinear) transformation from R^2 to R^2 given by

u=3y
v=3x^7-6y

So y= u/3. Put that into the second equation: v= 3x^7- 6(u/3)= 3x^7- 2u.
Solve that for x.

x=?
y=?

The Attempt at a Solution

- I'm really not sure what to do on this problem. We haven't seen any problems even similar to it in class, so I'm looking for help on it.

 

[snoggerT]

well, that was much easier than I thought it would be. Can you explain to me why that is the inverse?

 

[HallsofIvy]

What do you think an inverse is? You had u= 3y, v= 3x^7- 6y and you said the answer must be in the form x=, y= . I reduce the two equation to that form, solving for x and y.

Perhaps more specifically, if you start with (x, y) and apply the original tranform, you get (3y, 3x^7 - 6y). Now what happens if you apply the tranformation x= ((v+2u)/3)^(1/7, y= u/3? Since u= 3y, the second gives y= (3y)/3= y immediately. Since u= 3y and v= 3x^7- 6y, the x= ((3x^7- 6y+ 2(3y))/3)^(1/7)= ((3x^7/3)^(1/7)= (x^7)^(1/7)= x.
That's what an inverse is supposed to do.