document.write( "Question 1207307: Hi
\n" ); document.write( "Alan has 13 pieces of $2 $5 and $10 notes in his wallet. If the amount of money he has is $67 how many pieces of each note does he have \n" ); document.write( "
Algebra.Com's Answer #845098 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let the numbers of pieces of $2, $5, and $10 notes be, respectively,\r\n" );
document.write( "x, y, and z. \r\n" );
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document.write( "\"system%28x%2By%2Bz=13%2C2x%2B5y%2B10z=67%29\"\r\n" );
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document.write( "Multiply the first equation by -2\r\n" );
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document.write( "\"system%28-2x-2y-2z=-26%2C2x%2B5y%2B10z=67%29\"\r\n" );
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document.write( "Add the equations term by term:\r\n" );
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document.write( "\"3y%2B8z=41\"\r\n" );
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document.write( "Write all the integers in terms of their nearest multiple of the\r\n" );
document.write( "smallest absolute value of the coefficients of variables, \r\n" );
document.write( "which is 3:\r\n" );
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document.write( "\"3y%2B%289-1%29z=42-1\"\r\n" );
document.write( "\"3y%2B9z-z=42-1\"\r\n" );
document.write( "Divide thru by 3\r\n" );
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document.write( "\"y%2B3z-z%2F3=14-1%2F3\"\r\n" );
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document.write( "Isolate the fraction terms \r\n" );
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document.write( "\"y%2B3z-14=z%2F3-1%2F3\"\r\n" );
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document.write( "The left side is an integer, so the right side must also be an integer, \r\n" );
document.write( "say A\r\n" );
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document.write( "\"system%28y%2B3z-14=A%2Cz%2F3-1%2F3=A%29\"\r\n" );
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document.write( "Multiply the second equation by 3\r\n" );
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document.write( "\"system%28y%2B3z-14=A%2Cz-1=3A%29\"\r\n" );
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document.write( "\"z=3A%2B1\"\r\n" );
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document.write( "Substitute for z in y+3z-14=A:\r\n" );
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document.write( "\"y%2B3%283A%2B1%29-14=A\"\r\n" );
document.write( "\"y%2B9A%2B3-14=A\"\r\n" );
document.write( "\"y%2B9A-11=A\"\r\n" );
document.write( "\"y=11-8A\"\r\n" );
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document.write( "Substitute for y and z in\r\n" );
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document.write( "\"x%2By%2Bz=13\"\r\n" );
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document.write( "\"x%2B11-8A%2B3A%2B1=13\"\r\n" );
document.write( "\"x%2B12-5A=13\"\r\n" );
document.write( "\"x=5A%2B1\"\r\n" );
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document.write( "The general solution in integers is\r\n" );
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document.write( "\"system%28x=5A%2B1%2Cy=11-8A%2Cz=3A%2B1%29\"\r\n" );
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document.write( "x, y, and z must be non-negative integers:\r\n" );
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document.write( "\"system%285A%2B1%3E=0%2C11-8A%3E=0%2C3A%2B1%3E=0%29\" => \"system%28A%3E=-1%2F5%2CA%3C=11%2F8%2CA%3E=-1%2F3%29\"\r\n" );
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document.write( "Thus from the above, A can only be 0 or 1\r\n" );
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document.write( "Choose A=0\r\n" );
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document.write( "x=1, y=11, z=1\r\n" );
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document.write( "Choose A=1\r\n" );
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document.write( "x=6, y=3, z=4\r\n" );
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document.write( "Two solutions. [Other choices for A will produce one or more negative answers.]\r\n" );
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document.write( "Edwin
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