Normally in algebra, you can simplify an equation such as $a=a*b$ by dividing both sides by a common coefficient. For example, $\frac {a}{a}=\frac {ab}{a} \rightarrow1=b$. Obviously $b = 1$ is a correct solution to the equation, but clearly also is $a = 0$. Both solutions can be found inuitively, but it occurs to me that there should be a procedure that yields both, and I don't even know one that yields $0$ at all.
simplify $a=a*b$
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algebra-precalculus
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0When you divided by $a$ you implicitly assumed $a\ne 0$. It remains to consider the case $a=0$. – 2011-11-24
2 Answers
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$ a = a \ast b \Rightarrow a \cdot (1-b) =0$ which says either $b=1$ or $a=0$ because when the product of $2$ numbers is $0$, then either one of them has to be $0$.
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0Note the algebra-precalculus tag - in this setting, we can assume integral domains. – 2012-09-19
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This question has a good application of the zero product property.$\begin{aligned}a &=ab & \\ a - a b &= 0\\a(1 - b) & = 0 \\ \therefore a = 0 \vee b = 1\end{aligned} $
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0Good point... editing now. – 2012-09-19