Step 1 :Given the system of equations: \(u = -7x - 5y\) and \(v = -6x + 4y\)
Step 2 :Multiply the first equation by 4 and the second equation by 5 to get: \(4u = -28x - 20y\) and \(5v = -30x + 20y\)
Step 3 :Add these two equations together to get: \(4u + 5v = -58x\)
Step 4 :Solve for x to get: \(x = -\frac{4u + 5v}{58}\)
Step 5 :Substitute x into the first equation to get: \(u = -7x - 5y\) which simplifies to \(u = \frac{28u + 35v}{58} - 5y\)
Step 6 :Rearrange to get: \(58u = 28u + 35v - 290y\) which simplifies to \(30u = 35v - 290y\)
Step 7 :Solve for y to get: \(y = \frac{35v - 30u}{290}\)
Step 8 :The inverse transformations are: \(x = -\frac{4u + 5v}{58}\) and \(y = \frac{35v - 30u}{290}\)
Step 9 :Calculate the partial derivatives of x and y with respect to u and v to get: \(\frac{\partial x}{\partial u} = -\frac{4}{58}\), \(\frac{\partial x}{\partial v} = -\frac{5}{58}\), \(\frac{\partial y}{\partial u} = -\frac{30}{290}\), and \(\frac{\partial y}{\partial v} = \frac{35}{290}\)
Step 10 :The Jacobian is: \(\frac{\partial (x, y)}{\partial (u, v)} = \frac{\partial x}{\partial u} \cdot \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \cdot \frac{\partial y}{\partial u} = -\frac{140}{16820} + \frac{150}{16820} = \frac{10}{16820} = \frac{1}{1682}\)
Step 11 :\(\boxed{\text{So, the Jacobian of the transformation is } \frac{1}{1682}}\)