document.write( "Question 974863: find a 3-by-3 magic square using the numbers 6,7,8,14,15,16,22,23 and 24\r
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Algebra.Com's Answer #596696 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let the unknowns be A,B,C,D,E,F,G, like this \r\n" );
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document.write( "7 A B\r\n" );
document.write( "C D E\r\n" );
document.write( "F 8 G\r\n" );
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document.write( "I will number my equations in parentheses:\r\n" );
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document.write( "We sum all the numbers that go in the magic square:\r\n" );
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document.write( "(1) 6+7+8+14+15+16+22+23+24=135\r\n" );
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document.write( "therefore:\r\n" );
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document.write( "Each row, column, and diagonal must sum to 1/3 of 135 or 45\r\n" );
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document.write( "(2) 7+A+B = C+D+E = F+8+G = 7+C+F = A+D+8 = B+E+G = 7+D+G = B+D+F = 45\r\n" );
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document.write( "From (2) we get a system of 8 equations in 7 unknowns:\r\n" );
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document.write( "(3)  A + B                      = 38\r\n" );
document.write( "(4)          C + D + E          = 45\r\n" );
document.write( "(5)                       F + G = 37\r\n" );
document.write( "(6)          C          + F     = 38\r\n" );
document.write( "(7)  A         + D              = 37\r\n" );
document.write( "(8)     B          + E      + G = 45\r\n" );
document.write( "(9)              D          + G = 38\r\n" );
document.write( "(10)    B      + D      + F     = 45      \r\n" );
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document.write( "Eliminate A by subtracting (3)-(7), getting (11)\r\n" );
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document.write( "(4)          C + D + E          = 45\r\n" );
document.write( "(5)                       F + G = 37\r\n" );
document.write( "(6)          C          + F     = 38\r\n" );
document.write( "(8)     B          + E      + G = 45\r\n" );
document.write( "(9)              D          + G = 38\r\n" );
document.write( "(10)    B      + D      + F     = 45\r\n" );
document.write( "(11)    B      - D              =  1\r\n" );
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document.write( "Eliminate C by subtracting (4)-(6) getting (12)\r\n" );
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document.write( "(5)                       F + G = 37\r\n" );
document.write( "(8)     B          + E      + G = 45\r\n" );
document.write( "(9)              D          + G = 38\r\n" );
document.write( "(10)    B      + D      + F     = 45\r\n" );
document.write( "(11)    B      - D              =  1\r\n" );
document.write( "(12)             D + E  - F     =  7\r\n" );
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document.write( "Eliminate E by subtracting (8)-(12) getting (13)\r\n" );
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document.write( "(5)                       F + G = 37\r\n" );
document.write( "(9)              D          + G = 38\r\n" );
document.write( "(10)    B      + D      + F     = 45\r\n" );
document.write( "(11)    B      - D              =  1\r\n" );
document.write( "(13)    B      - D      + F + G = 38\r\n" );
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document.write( "Eliminate D by adding (9)+(10)+(11)+(13) getting (14)\r\n" );
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document.write( "(5)                      F +  G  =  37\r\n" );
document.write( "(14)   3B             + 2F + 2G  = 122\r\n" );
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document.write( "We eliminate F and G by multiplying (5) by -2 getting (15)\r\n" );
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document.write( "(15)                 - 2F - 2G  =  -74 \r\n" );
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document.write( "and adding to (14) \r\n" );
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document.write( "(15)                  - 2F - 2G  = -74\r\n" );
document.write( "(14)   3B             + 2F + 2G  = 122\r\n" );
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document.write( "       3B                        =  48 \r\n" );
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document.write( "so B = 16 \r\n" );
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document.write( "Substituting in (11)           Substituting in (3)\r\n" );
document.write( "(11)    B - D = 1                      A + B = 38\r\n" );
document.write( "       16 - D = 1                     A + 16 = 38 \r\n" );
document.write( "           -D = -15                        A = 22\r\n" );
document.write( "            D = 15\r\n" );
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document.write( "Substituting in (9)\r\n" );
document.write( "(9)      D + G = 38\r\n" );
document.write( "        15 + G = 38\r\n" );
document.write( "             G = 23\r\n" );
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document.write( "Substituting in (5)\r\n" );
document.write( "(5)     F + G = 37\r\n" );
document.write( "       F + 23 = 37\r\n" );
document.write( "            F = 14\r\n" );
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document.write( "Substituting in (6)\r\n" );
document.write( "(6)     C + F = 38\r\n" );
document.write( "       C + 14 = 38\r\n" );
document.write( "            C = 24\r\n" );
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document.write( "Substituting C = 24 and D = 15 in (4)\r\n" );
document.write( "(4)  C + D + E  = 45\r\n" );
document.write( "   24 + 15 + E = 45 \r\n" );
document.write( "        39 + E = 45\r\n" );
document.write( "             E =  6 \r\n" );
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document.write( "So the magic square\r\n" );
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document.write( "7 A B     7 22 16\r\n" );
document.write( "C D E  = 24 15  6 \r\n" );
document.write( "F 8 G    14  8 23\r\n" );
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document.write( "Edwin
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