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Find the number of ordered pairs (m,n) which satisfy the system 4(m-n) = 7(1-n), 3/2n = 4-2m

 Oct 1, 2022
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the number of ordered pairs (m,n)

 

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\(\color{blue}3/2n = 4-2m\\ 2n=\dfrac{3}{4-2m}\\ n=\dfrac{3}{8-4m}\\ \color{blue}4(m-n) = 7(1-n)\\ 4m-\dfrac{12}{8-4m}= 7-\dfrac{21}{8-4m}\\ \dfrac{32m-16m^2-12}{8-4m}=\dfrac{56-28m-21}{8-4m}\)

\(-16m^2+32m+28m-12-56+21=0\\ 16m^2-60m+47=0\\ \text{a_______b_____c}\\ m = {-b \pm \sqrt{b^2-4ac} \over 2a}\\ m\in\{\frac{15-\sqrt{37}}{8},\frac{15+\sqrt{37}}{8}\}\\ \color{blue}m\in \{1.11465,2.63534\}\)

\(n=\frac{3}{8-4m}\\ n\in \{\frac{3}{8-4\cdot \frac{15-\sqrt{37}}{8}},\frac{3}{8-4\cdot \frac{15+\sqrt{37}}{8}}\}\\ \color{blue}n\in \{0.847127,-1.18046\}\)

laugh  !

 Oct 1, 2022
edited by asinus  Oct 1, 2022
edited by asinus  Oct 1, 2022
edited by asinus  Oct 1, 2022

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