document.write( "Question 549740: Please help me solve the following system for (x,y):
\n" ); document.write( "\"log%289%2Cx%29\" + \"log%28y%2C8%29\" = 2
\n" ); document.write( "\"log%28x%2C9%29\" + \"log%288%2Cy%29\" = \"8%2F3\"\r
\n" ); document.write( "\n" ); document.write( "thanks \n" ); document.write( "
Algebra.Com's Answer #358152 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\"log%289%2Cx%29\" + \"log%28y%2C8%29\" = 2
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document.write( "Let u = \"log%289%2Cx%29\".\r\n" );
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document.write( "Then use the rule of logarithms \"log%28B%2CA%29=1%2Flog%28A%2CB%29\" to write \r\n" );
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document.write( "\"log%28x%2C9%29\" = \"1%2Flog%289%2Cx%29\" = \"1%2Fu\"\r\n" );
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document.write( "Let v = \"log%288%2Cy%29\".\r\n" );
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document.write( "Then use the same rule of logarithms \"log%28B%2CA%29=1%2Flog%28A%2CB%29\" to write \r\n" );
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document.write( "\"log%28y%2C8%29\" = \"1%2Flog%288%2Cy%29\" = \"1%2Fv\"\r\n" );
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document.write( "So now the system is\r\n" );
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document.write( "u + \"1%2Fv\" = 2\r\n" );
document.write( "\"1%2Fu\" + v = \"8%2F3\"\r\n" );
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document.write( "Clear each of fractions by multiplying the first thru by v and the\r\n" );
document.write( "second one through by 3u\r\n" );
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document.write( " uv + 1 = 2v\r\n" );
document.write( "3 + 3uv = 8u\r\n" );
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document.write( "Solve the first for uv\r\n" );
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document.write( " uv + 1 = 2v\r\n" );
document.write( "     uv = 2v - 1\r\n" );
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document.write( "Substitute for uv in the second:\r\n" );
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document.write( "3 + 3uv = 8u\r\n" );
document.write( "3 + 3(2v-1) = 8u\r\n" );
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document.write( "3 + 6v - 3 = 8u\r\n" );
document.write( "        6v = 8u\r\n" );
document.write( "        3v = 4u\r\n" );
document.write( "         v = \"%284u%29%2F3\"\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "3 + 3uv = 8u\r\n" );
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document.write( "3 + 3u\"%28%284u%29%2F3%29\" = 8u\r\n" );
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document.write( "Cancel 3's in the second term:\r\n" );
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document.write( "3 + u(4u) = 8u\r\n" );
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document.write( "3 + 4u² = 8u\r\n" );
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document.write( "    4u² - 8u + 3 = 0\r\n" );
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document.write( "(2u - 3)(2u - 1) = 0\r\n" );
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document.write( "2u - 3 = 0;      2u - 1 = 0\r\n" );
document.write( "    2u = 3;          2u = 1\r\n" );
document.write( "     u = \"3%2F2\"            u = \"1%2F2\"\r\n" );
document.write( "     \r\n" );
document.write( "    3v = 4u          3v = 4u\r\n" );
document.write( "    3v = 4\"%283%2F2%29\"        3v = 4\"%281%2F2%29\" \r\n" );
document.write( "    3v = 6           3v = 2\r\n" );
document.write( "     v = 2            v = \"2%2F3\"       \r\n" );
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document.write( "So we have (u,v) = (\"3%2F2\",2) or (u,v) = (\"1%2F2\",\"2%2F3\")\r\n" );
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document.write( "But we want x and y, not u and v, so we substitute back in each case:\r\n" );
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document.write( "(u,v) = (\"3%2F2\",2)\r\n" );
document.write( "        \r\n" );
document.write( " u = \"log%289%2Cx%29\"         v = \"log%288%2Cy%29\"\r\n" );
document.write( " \"3%2F2\" = \"log%289%2Cx%29\"         2 = \"log%288%2Cy%29\"    \r\n" );
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document.write( "Write in exponential form using the rule: \"A=log%28B%2CC%29\" is equivalent to \"C=B%5EA\"\r\n" );
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document.write( "x = \"matrix%282%2C1%2C%22%22%2C9%5E%283%2F2%29%29\" = \"%28sqrt%289%29%29%5E3\" = 3³ = 27        y = 8² = 64\r\n" );
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document.write( "So one solution is (x,y) = (27,64)\r\n" );
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document.write( "For the other case:\r\n" );
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document.write( "(u,v) = (\"1%2F2\",\"2%2F3\")\r\n" );
document.write( "        \r\n" );
document.write( " u = \"log%289%2Cx%29\"         v = \"log%288%2Cy%29\"\r\n" );
document.write( " \"1%2F2\" = \"log%289%2Cx%29\"         \"2%2F3\" = \"log%288%2Cy%29\"    \r\n" );
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document.write( "Write in exponential form using the rule: \"A=log%28B%2CC%29\" is equivalent to \"C=B%5EA\"\r\n" );
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document.write( "x = \"matrix%282%2C1%2C%22%22%2C9%5E%281%2F2%29%29\" = \"sqrt%289%29\" = 3        y = \"matrix%282%2C1%2C%22%22%2C8%5E%282%2F3%29%29\" = \"%28root%283%2C8%29%29%5E2\" = 2² = 4\r\n" );
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document.write( "So the other solution is (x,y) = (3,4)\r\n" );
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document.write( "There are two solutions:\r\n" );
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document.write( "(x,y) = (27,64)  and   (x,y) = (3,4)\r\n" );
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document.write( "Edwin
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