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three linear equations with a nonlinear objective

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November 15, 2009
linear algebramatrix

Problem Statement

Let a a, b b, c c, x x, y y, and z z be real numbers that satisfy the three equations \begin{align*} 13x + by + cz &= 0 \\ ax + 23y + cz &= 0 \\ ax + by + 42z &= 0. \end{align*}Suppose that a≠13 a \ne 13 and x≠0 x \ne 0. What is the value of 13a−13+23b−23+42c−42 ? \frac{13}{a - 13} + \frac{23}{b - 23} + \frac{42}{c - 42} \, ?
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