Solve the system u=2x-y v= 3 x+2y, for x and y in terms of u and v then find the value of Jacobian σ(x, y)/σ(u, v). Find the image under the transformation of the triangular region with the vertices (0,0) ,(2,4), and (2, -3) in the x, y plane. Sketch the transformed region in uv plane. The function for x in terms of u and v is x=?

x =

y =  

Jacobian value = 1/7

Step-by-step explanation:

Given data:

u = 2x -y

v = 3x + 2y

solving the system for  x and  y in terms of u and v

x =

y =  

The value of the Jacobian =  1/7 ( solution attached below )

Image under the transformation of the triangular region with vertices ( 0,0) , (2,4) and (2,-3) in the x-y plane is attached below

{x = -1/2 ,   y = -1

step-by-step explanation:

solve the following system:

{2 x + y = -2

{10 x - 3 y = -2

in the first equation, look to solve for y:

{2 x + y = -2

{10 x - 3 y = -2

subtract 2 x from both sides:

{y = -2 x - 2

{10 x - 3 y = -2

substitute y = -2 x - 2 into the second equation:

{y = -2 x - 2

{10 x - 3 (-2 x - 2) = -2

10 x - 3 (-2 x - 2) = (6 x + 6) + 10 x = 16 x + 6:

{y = -2 x - 2

{16 x + 6 = -2

in the second equation, look to solve for x:

{y = -2 x - 2

{16 x + 6 = -2

subtract 6 from both sides:

{y = -2 x - 2

{16 x = -8

divide both sides by 16:

{y = -2 x - 2

{x = -1/2

substitute x = -1/2 into the first equation:

{y = -1

{x = -1/2

collect results in alphabetical order:

answer:   {x = -1/2 ,   y = -1

the answer is y = 11.54

hope this : )