For supplemental problem 2, G-2 : http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-a-functions-of-two-variables-tangent-approximation-and-optimization/problem-set-4/MIT18_02SC_SupProb2.pdf ,
Isn't it enough to simply show the equation always has solutions as long as the determinant is not zero? And that if all the xi's are the same, the determinant will be zero, and thus no solution (in a vertical line)?
In other words, what does the Hint mean? How do we apply the Hint to solve the problem?

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*so what does the hint mean? Isn't all that unnecessary?

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