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The problem in the image provides a scenario where \( u, v, w \) are the roots of a given equation, and we need to determine the Jacobian determinant:

\[
\frac{\partial(x, y, z)}{\partial(u, v, w)}.
\]

Given the equation provided:
\[
\frac{x}{a + \lambda} + \frac{y}{b + \lambda} + \frac{z}{c + \lambda} = 1,
\]
and knowing that \( u, v, w \) are the roots in terms of \( \lambda \), the expression to be found is the Jacobian of the transformation from the variables \( u, v, w \) to the variables \( x, y, z \).

The answer provided in the image is:
\[
\frac{(u - v)(v - w)(w - u)}{(a - b)(b - c)(c - a)}.
\]

This Jacobian essentially represents how the variables \( x, y, z \) change with respect to the roots \( u, v, w \), and the answer shows that the determinant of this transformation involves the differences between the roots and the constants \( a, b, c \).

Would you like further details on the step-by-step derivation of this result or additional clarification? 

Here are some related questions to extend the understanding:

1. How are the roots \( u, v, w \) related to the original equation?
2. What does the Jacobian determinant represent geometrically in this context?
3. How can we compute the partial derivatives involved in the Jacobian?
4. What role do the constants \( a, b, c \) play in the equation?
5. How does the form of the answer relate to standard results in systems of equations?

**Tip**: The Jacobian determinant often reflects how volumes (or areas in 2D) transform under a change of variables.

If u, v, w are the roots of the equation in λ and x/(a + λ) + y/(b + λ) + z/(c + λ) = 1, then find ∂(x, y, z)/∂(u, v, w).

This problem explores the Jacobian determinant involving the transformation of variables x, y, z with respect to the roots u, v, w. Solve the equation x/(a + λ) + y/(b + λ) + z/(c + λ) = 1 and find the Jacobian ∂(x, y, z)/∂(u, v, w) with a detailed solution involving differences between the roots and constants.

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