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- Substitution of variables can be used to reduce two equations in two unknowns to a single equation in one of them.
If in a problem the value of a certain variable is neither given nor requested, a reasonable approach is to eliminate that variable.

- x
^{a}x^{b}= x^{a + b}

x^{a}/ x^{b}= x^{a - b}

(x^{a})^{b}= x^{a b} - If A x
^{2}+ B x + C = 0, x = (-B ± √ (B^{2}- 4 A C)) / (2 A). - e
^{ln x}= x

ln (e^{x}) = x - The graph of y = a f(x + b) is the graph of f(x), dilated in the y direction by the factor a and translated to the left by b.
- n! ≡ n * (n - 1) * (n -2) * ... * 3 * 2 * 1
- The number of ways of choosing m items from a set of n items, when order does
*not*matter, isn! / (m! (n-m)!)

- The weighted average of a quantity Q with values in two sub-populations (a and b) is
<Q> = f

where f_{a}* Q_{a}+ (1 - f_{a}) * Q_{b},_{a}is the fraction of the total population having value Q_{a}, and the rest of the population has value Q_{b}.

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©2012, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.