document.write( "Question 141333: Please help me solve this simultaneous equation:\r
\n" ); document.write( "\n" ); document.write( "log x base 3 = y = log (2x-1) base 9\r
\n" ); document.write( "\n" ); document.write( "and another one:\r
\n" ); document.write( "\n" ); document.write( "log (x+y) = 0\r
\n" ); document.write( "\n" ); document.write( "2 log x = log (y+1) \n" ); document.write( "
Algebra.Com's Answer #102975 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Please help me solve this simultaneous equation:\r
\n" ); document.write( "\n" ); document.write( "\"log%283%2Cx%29=y=log%289%2C%282x-1%29%29\"
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\r\n" );
document.write( "Write two separate equations:\r\n" );
document.write( "\r\n" );
document.write( "\"y=log%283%2Cx%29\"\r\n" );
document.write( "\"y=log%289%2C%282x-1%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Use this principle to rewrite each: \r\n" );
document.write( "                \"A=log%28B%2CC%29\" can be rewritten as \"C=B%5EA\"\r\n" );
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document.write( "Rewriting the first:\r\n" );
document.write( "\"x=3%5Ey\"\r\n" );
document.write( "\r\n" );
document.write( "Rewriting the second:\r\n" );
document.write( "\"2x-1=9%5Ey\"\r\n" );
document.write( "\r\n" );
document.write( "Now rewrite \"9\" as \"%283%2A3%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"2x-1=%283%2A3%29%5Ey\"\r\n" );
document.write( "\r\n" );
document.write( "or, multiplying exponents:\r\n" );
document.write( "\r\n" );
document.write( "\"2x-1=%283%5Ey%29%283%5Ey%29\"\r\n" );
document.write( "\r\n" );
document.write( "So now we have this system:\r\n" );
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document.write( "\"2x-1=%283%5Ey%29%283%5Ey%29\"\r\n" );
document.write( "\"x=3%5Ey\"\r\n" );
document.write( "\r\n" );
document.write( "Using the second, we can substitute\r\n" );
document.write( "\"x\" for \"3%5Ey\" in the first:\r\n" );
document.write( "\r\n" );
document.write( "\"2x-1=%28x%29%28x%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"2x-1=x%5E2\"\r\n" );
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document.write( "Get 0 on the left\r\n" );
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document.write( "\"0=x%5E2-2x%2B1\"\r\n" );
document.write( "\r\n" );
document.write( "Factor the right side:\r\n" );
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document.write( "\"0=%28x-1%29%28x-1%29\"\r\n" );
document.write( "\r\n" );
document.write( "Using the zero factor principle,\r\n" );
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document.write( "\"x-1=0\" gives \"x=1\"\r\n" );
document.write( "\"x-1=0\" gives \"x=1\"\r\n" );
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document.write( "So there is only one value for \"x\",\r\n" );
document.write( "which is \"1\"\r\n" );
document.write( "\r\n" );
document.write( "Substitute \"1\" for x in \"x=3%5Ey\"\r\n" );
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document.write( "\"1=3%5Ey\"\r\n" );
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document.write( "Now we use the rule:\r\n" );
document.write( "            Any number raised to the zero power, \r\n" );
document.write( "            except zero itself, equals 1\r\n" );
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document.write( "Thus \"y=0\"\r\n" );
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document.write( "and the solution is \"x=1\", \"y=0\"\r\n" );
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document.write( "Let's check:\r\n" );
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document.write( "\"log%283%2Cx%29=y=log%289%2C%282x-1%29%29\"\r\n" );
document.write( "\"log%283%2C%281%29%29=0=log%289%2C%282%281%29-1%29%29\"\r\n" );
document.write( "\"log%283%2C%281%29%29=0=log%289%2C%282-1%29%29\"\r\n" );
document.write( "\"log%283%2C%281%29%29=0=log%289%2C%281%29%29\"\r\n" );
document.write( "\"log%283%2C%281%29%29=0\" can be rewritten as \"3%5E0=1\" which is true.\r\n" );
document.write( "\"log%289%2C%281%29%29=0\" can be rewritten as \"9%5E0=1\" which is also true.\r\n" );
document.write( "\r\n" );
document.write( "-------------------------------\r\n" );
document.write( "\r\n" );
document.write( "and another one:\r\n" );
document.write( "\r\n" );
document.write( "\"log%28%28x%2By%29%29+=+0\"\r\n" );
document.write( "\"2%2Alog%28x%29+=+log+%28%28y%2B1%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "When no base is written, the base \"10\" is understood:\r\n" );
document.write( "\r\n" );
document.write( "\"log%2810%2C%28x%2By%29%29+=+0\"\r\n" );
document.write( "\"2%2Alog%2810%2C%28x%29%29+=+log%28+10%2C%28y%2B1%29+%29\"\r\n" );
document.write( "\r\n" );
document.write( "We can rewrite the first using the rule:\r\n" );
document.write( "         \"log%28B%2CA%29=C\" can be rewritten as \"A=B%5EC\"\r\n" );
document.write( "         \r\n" );
document.write( "\"log%2810%2C%28x%2By%29%29+=+0\"\r\n" );
document.write( "becomes\r\n" );
document.write( "\"x%2By+=+10%5E0\"\r\n" );
document.write( "\"x%2By=1\"\r\n" );
document.write( "\r\n" );
document.write( "We can use this rule on the left side of the second eq:\r\n" );
document.write( "                      \"A%2Alog%28B%2C%28C%29%29=log%28B%2C%28C%5EA%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"2%2Alog%2810%2C%28x%29%29+=+log%28+10%2C%28y%2B1%29+%29\"\r\n" );
document.write( "becomes\r\n" );
document.write( "\"log%2810%2C%28x%5E2%29%29+=+log%28+10%2C%28y%2B1%29+%29\"\r\n" );
document.write( "\r\n" );
document.write( "Now we use the principle: \r\n" );
document.write( "                \"log%28B%2CA%29=log%28B%2CC%29\" can be rewritten \"A=C\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2=y%2B1\"\r\n" );
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document.write( "So we have the system of ewquations:\r\n" );
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document.write( "\"x%2By=1\"\r\n" );
document.write( "\"x%5E2=y%2B1\"\r\n" );
document.write( "          \r\n" );
document.write( "Can you solve that system of equations by substitution?\r\n" );
document.write( "If not post again asking how.\r\n" );
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document.write( "That last system of equations has two ordered pairs \r\n" );
document.write( "of solutions:\r\n" );
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document.write( "(\"x\",\"y\")= (\"1\",\"0\") and (\"x\",\"y\") = (\"-2\",\"3\")\r\n" );
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document.write( "However we must check them, because sometimes a solution\r\n" );
document.write( "to our final equations is not a solution to the original\r\n" );
document.write( "equation:\r\n" );
document.write( "\r\n" );
document.write( "Checking (\"x\",\"y\")= (\"1\",\"0\") in the first equation:\r\n" );
document.write( "\r\n" );
document.write( "\"log%2810%2C%281%2B0%29%29+=+0\"\r\n" );
document.write( "\"log%2810%2C%281%29%29+=+0\"\r\n" );
document.write( "\"10%5E0=1\"\r\n" );
document.write( "\"1=1\"\r\n" );
document.write( "\r\n" );
document.write( "Checking (\"x\",\"y\")= (\"1\",\"0\") in the second equation:\r\n" );
document.write( "\r\n" );
document.write( "\"2%2Alog%2810%2C%281%29%29+=+log%28+10%2C%280%2B1%29+%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"2%2Alog%2810%2C1%29+=+log%28+10%2C%280%2B1%29+%29\"\r\n" );
document.write( "\"2%2Alog%2810%2C1%29+=+log%28+10%2C%281%29+%29\"\r\n" );
document.write( "Since \"log%2810%2C1%29=0\"\r\n" );
document.write( "\"2%2A0=0\"\r\n" );
document.write( "\"0=0\"\r\n" );
document.write( "\r\n" );
document.write( "----\r\n" );
document.write( "\r\n" );
document.write( "Checking (\"x\",\"y\")= (\"-2\",\"3\") in the first equation:\r\n" );
document.write( "\r\n" );
document.write( "\"log%2810%2C%28-2%2B3%29%29+=+0\"\r\n" );
document.write( "\"log%2810%2C%281%29%29+=+0\"\r\n" );
document.write( "\"10%5E0=1\"\r\n" );
document.write( "\"1=1\"\r\n" );
document.write( "\r\n" );
document.write( "Checking (\"x\",\"y\")= (\"-2\",\"3\") in the second equation:\r\n" );
document.write( "\r\n" );
document.write( "\"2%2Alog%2810%2C%28-2%29%29+=+log%28+10%2C%283-2%29+%29\"\r\n" );
document.write( "\r\n" );
document.write( "We can stop here because the first term is \r\n" );
document.write( "undefined because the logarithm of a negative \r\n" );
document.write( "number is not defined (except in certain \r\n" );
document.write( "advanced mathematics, but never in\r\n" );
document.write( "ordinary algebra)\r\n" );
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document.write( "So there is but one solution, (\"x\",\"y\")= (\"1\",\"0\")\r\n" );
document.write( "\r\n" );
document.write( "Edwin
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