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Find all values $$a$$ for which there exists an ordered pair $$(a,b)$$ satisfying the following system of equations: \begin{align*} a + ab^2 & = 40b, \\ a - ab^2 & = -32b. \end{align*}

List only the values for $$a$$.

Feb 18, 2019

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$$a+ab^2 = 40b\\ a-ab^2 = -32b\\$$

$$\text{add them}\\ 2a=8b\\ b=\dfrac{a}{4}$$

$$\text{subtract them}\\ 2ab^2 = 72b\\ \text{if }b\neq 0\\ 2ab=72\\ 2\dfrac{a^2}{4} = 72\\ a^2 = 144\\ a=\pm 12$$

$$\text{and of course }a=b=0 \text{ is also a solution}\\ \text{so }a = 0, \pm 12$$

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Feb 18, 2019