$$3 − 1 · (23 − 7 · 3) = −1 · 23 + 8 · 3$$
How does one get the left side to become the right side? Is it algebra? My discrete math textbook just wrote this but never explained a step by step process of how it came to be.
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1Just expand: $3 - 1 \cdot (23 - 7 \cdot 3) = 3 - 1 \cdot 23 + 7 \cdot 3 = -1 \cdot 23 + 8 \cdot 3$. – TMM Mar 16 '14 at 21:47
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If you use the version of the extended Euclidean algorithm linked in this answer it will all be much simpler. – Bill Dubuque Mar 16 '14 at 22:55
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To get from the left side expression to the right, treating $\color{blue}{s = 23}$ and $\color{red}{t = 3}$, $$\begin{aligned} \color{red}3 - 1 \cdot (\color{blue}{23} - 7 \cdot \color{red}3) &= \color{red}t - 1 \cdot (\color{blue}s - 7 \cdot \color{red}t)\\ &= \color{red}t - \color{blue}s + 7\color{red}t\\ &= 8\color{red}t - \color{blue}s\\ &= -\color{blue}s + 8\color{red}t\\ &= -1 \cdot \color{blue}{23} + 8 \cdot \color{red}3 \end{aligned}$$
This is how one obtains this type of equivalence without directly simplifying the whole expression.
Mr Mathster
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