document.write( "Question 1041233: I shall write a two digit number sum of two digit number is 14 and if 29 is subtracted from the number two digits will be equal .Led us from the simultaneous equations by solving them let us see what will be the two digit number. \n" ); document.write( "
Algebra.Com's Answer #656259 by Edwin McCravy(20056) $\"\"$ $\"About$
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The other tutor has made the error of assuming that the\r\n" );
document.write( "number with the equal digits must have both its digits\r\n" );
document.write( "as the same digit as the units digit of the desired number.\r\n" );
document.write( "This is not a requirement.\r\n" );
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document.write( "Let a = the tens digit\r\n" );
document.write( "Let b = the units digit\r\n" );
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document.write( "The number = 10a + b\r\n" );
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document.write( "Since the sum of the digits is 14,\r\n" );
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document.write( "a + b = 14, or b = 14 - a\r\n" );
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document.write( "Let c = the digit that is the same in the\r\n" );
document.write( "answer when 29 is subtracted from the number.\r\n" );
document.write( "So the answer when 29 is subtracted from the\r\n" );
document.write( "number is 10c + c or 11c  \r\n" );
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document.write( "10a + b - 29 = 11c\r\n" );
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document.write( "So the system of equations is\r\n" );
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document.write( "This is underdetermined because there are more\r\n" );
document.write( "unknowns than equations.\r\n" );
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document.write( "We substitute 14 - a for b\r\n" );
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document.write( "10a + (14 - a) - 29 = 11c\r\n" );
document.write( "10a + 14 - a - 29 = 11c\r\n" );
document.write( "9a - 15 = 11c\r\n" );
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document.write( "9 is the smallest number in that equation in \r\n" );
document.write( "absolute value.  So write 15 and 11 in terms \r\n" );
document.write( "of their nearest multiple of 9.  So we write \r\n" );
document.write( "15 as 18 - 3 and 11 as 9 + 2\r\n" );
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document.write( "9a - (18 - 3) = (9 + 2)c\r\n" );
document.write( "9a - 18 + 3 = 9c + 2c\r\n" );
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document.write( "Divide through by 9\r\n" );
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document.write( "a - 2 + 3/9 = c + 2c/9\r\n" );
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document.write( "Get the fractions on one side, \r\n" );
document.write( "other terms on the other side:\r\n" );
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document.write( "2c/9 - 3/9 = a - 2 - c \r\n" );
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document.write( "The right side is an integer, so the left\r\n" );
document.write( "side is too.  Let that integer be I.\r\n" );
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document.write( "2c/9 - 3/9 = I    a - 2 - c = I; \r\n" );
document.write( "    2c - 3 = 9I\r\n" );
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document.write( "2 is the smallest number in that equation in \r\n" );
document.write( "absolute value.  So write 3 and 9 in terms \r\n" );
document.write( "of their nearest multiple of 2.  So we write \r\n" );
document.write( "3 as 2+1 and 9 as 8+1\r\n" );
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document.write( "    2c - 3 = 9I\r\n" );
document.write( "    2c - (2 + 1) = (8 + 1)I\r\n" );
document.write( "    2c - 2 - 1 = 8I + I\r\n" );
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document.write( "Divide through by 2\r\n" );
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document.write( "    c - 1 - 1/2 = 4I + I/2\r\n" );
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document.write( "Get the fractions on one side, \r\n" );
document.write( "other terms on the other side:\r\n" );
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document.write( "c - 1 - 4I = I/2 + 1/2\r\n" );
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document.write( "The left side is an integer, so the\r\n" );
document.write( "right side is too.  Let that integer be J.\r\n" );
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document.write( "c - 1 - 4I = J;    I/2 + 1/2 = J\r\n" );
document.write( "                         I+1 = 2J\r\n" );
document.write( "                           I = 2J - 1\r\n" );
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document.write( "Substitute for I\r\n" );
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document.write( "c - 1 - 4(2J - 1) = J\r\n" );
document.write( "   c - 1 - 8J + 4 = J\r\n" );
document.write( "       c - 8J + 3 = J\r\n" );
document.write( "                c = 9J - 3\r\n" );
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document.write( "The only value of J that will permit c to be\r\n" );
document.write( "a digit is J = 1\r\n" );
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document.write( "So c = 9J - 3 = 9(1) - 3 = 6\r\n" );
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document.write( "And since I = 2J - 1,\r\n" );
document.write( "          I = 2(1) - 1, = 2 - 1 = 1\r\n" );
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document.write( "Now the system of equations becomes:\r\n" );
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document.write( "Since a - 2 - c = I\r\n" );
document.write( "      a - 2 - 6 = 1\r\n" );
document.write( "          a - 8 = 1\r\n" );
document.write( "              a = 9\r\n" );
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document.write( "Since b = 14 - a\r\n" );
document.write( "      b = 14 - 9 = 5\r\n" );
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document.write( "So the number is 95\r\n" );
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document.write( "Checking:\r\n" );
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document.write( "           95\r\n" );
document.write( "          -29\r\n" );
document.write( "           66\r\n" );
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document.write( "Edwin
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