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SMUP 2010

Solusi SMUP 2010 Kode 041 – Nomor 25


Nilai-nilai\ x\ yang\ memenuhi\ persamaan
3^{2(9x-y-2)}-10.3^{9x-y-1}+81=0\ dan\ 8x-y+4=0
adalah\ ....
A.\quad  9\:\: atau\:\: -33\\B.\quad 33\:\: atau\:\: 9\\C.\quad -7\:\: atau\:\: -9\\D.\quad 7\:\: atau\:\: 9\\E.\quad -7\:\: atau\:\: 9
Jawaban: D

\begin{array}{rcl}3^{2(9x-y-2)}-10.3^{9x-y-1}+81&=&0\\\\3^{2(9x-y-1-1)}-10.3^{9x-y-1}+81&=&0\\\\3^{2(9x-y-1)-2}-10.3^{9x-y-1}+81&=&0\\\\3^{2(9x-y-1)}.3^{-2}-10.3^{9x-y-1}+81&=&0\\\\3^{2(9x-y-1)}-90.3^{9x-y-1}+729&=&0\end{array}
Misalkan\ p=3^{9x-y-1}
\begin{array}{rcl}p^2-90p+729&=&0\\\\(p-9)(p-81)&=&0\\\\p=9&atau&p=81\\\\3^{9x-y-1}=3^2&atau&3^{9x-y-1}=3^4\\\\9x-y-1=2&atau&9x-y-1=4\\\\9x-y=3&atau&9x-y=5\end{array}
Selisih\ dari\ 9x-y=3\ dan\ 8x-y=-4\ menghasilkan\ x=7
Selisih\ dari\ 9x-y=5\ dan\ 8x-y=-4\ menghasilkan\ x=9

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