document.write( "Question 519837: how would you solve this 4x4 system of equations:
\n" ); document.write( " 4w+ x+2y-3z=-16
\n" ); document.write( "-3w+3x- y+4z= 20
\n" ); document.write( " -w+2x+5y+ z= -4
\n" ); document.write( " 5w+4x+3y- z=-10 \n" ); document.write( "
Algebra.Com's Answer #345820 by Edwin McCravy(20056) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "(1)      4w +  x + 2y - 3z = -16\r\n" );
document.write( "(2)     -3w + 3x -  y + 4z =  20\r\n" );
document.write( "(3)      -w + 2x + 5y +  z =  -4\r\n" );
document.write( "(4)      5w + 4x + 3y -  z = -10\r\n" );
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document.write( "Pick a letter to eliminate.  I pick z.\r\n" );
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document.write( "Add equations (3) and (4) as they are\r\n" );
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document.write( "(3)      -w + 2x + 5y +  z =  -4\r\n" );
document.write( "(4)      5w + 4x + 3y -  z = -10\r\n" );
document.write( "--------------------------------\r\n" );
document.write( "         4w + 6x + 8y      = -14\r\n" );
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document.write( "That can be simplified by dividing through by 2\r\n" );
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document.write( "(5)     2w + 3x + 4y      =  -7\r\n" );
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document.write( "Multiply (4) by 4 [to add to (2) to eliminate z]\r\n" );
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document.write( "        20w + 16x + 12y - 4z = -40\r\n" );
document.write( "(2)     -3w +  3x -   y + 4z =  20\r\n" );
document.write( "----------------------------------\r\n" );
document.write( "(6)     17w + 19x + 11y      = -20       \r\n" );
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document.write( "Multiply (3) by 3 [to add to (1) to eliminate z]\r\n" );
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document.write( "        -3w + 6x + 15y + 3z =  -12\r\n" );
document.write( "(1)      4w +  x +  2y - 3z =  -16\r\n" );
document.write( "----------------------------------\r\n" );
document.write( "(7)       w + 7x + 17y      =  -28\r\n" );
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document.write( "Now we have reduced the system to 3 equations in 3 unknowns:\r\n" );
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document.write( "(5)      2w +  3x +  4y  =  -7\r\n" );
document.write( "(6)     17w + 19x + 11y  = -20\r\n" );
document.write( "(7)       w +  7x + 17y  = -28\r\n" );
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document.write( "Pick a letter to eliminate.  I pick w.\r\n" );
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document.write( "Multiply (7) by -2 [to add to (5) to eliminate w]\r\n" );
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document.write( "        -2w - 14x - 34y  =  56\r\n" );
document.write( "(5)      2w +  3x +  4y  =  -7\r\n" );
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document.write( "(8)           -11x - 30y  =  49\r\n" );
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document.write( "Multiply (7) by -17 [to add to (6) to eliminate w]\r\n" );
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document.write( "       -17w - 119x - 289y  = 476\r\n" );
document.write( "(6)     17w +  19x +  11y  = -20\r\n" );
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document.write( "             -100x - 278y  = 456\r\n" );
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document.write( "That can be simplified by dividing through by 2\r\n" );
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document.write( "(9)           -50x - 139y  = 228\r\n" );
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document.write( "Now we have reduced the system to 2 equations in 2 unknowns:\r\n" );
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document.write( "(8)           -11x -  30y  =  49\r\n" );
document.write( "(9)           -50x - 139y  = 228\r\n" );
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document.write( "Pick a letter to eliminate.  I pick x.\r\n" );
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document.write( "the least common multiple of 11 and 50 is 550. So we multiply\r\n" );
document.write( "(8) by 50, and(9) by -11 so that the x's will cancel:\r\n" );
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document.write( "            -550x - 1500y  =  2450\r\n" );
document.write( "             550x + 1529y  = -2508\r\n" );
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document.write( "                      29y  =   -58\r\n" );
document.write( "(10)                    y = -2\r\n" );
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document.write( "Substitute in (8)\r\n" );
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document.write( "           -11x -  30(-2)  =  49\r\n" );
document.write( "                 -11x + 60 =  49\r\n" );
document.write( "                      -11x = -11\r\n" );
document.write( "(11)                     x = 1\r\n" );
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document.write( "Substitute x = 1 and y = -2 in (5)\r\n" );
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document.write( "           2w +  3x +  4y  = -7\r\n" );
document.write( "         2w + 3(1) + 4(-2) = -7\r\n" );
document.write( "                2w + 3 - 8 = -7\r\n" );
document.write( "                    2w - 5 = -7\r\n" );
document.write( "                        2w = -2\r\n" );
document.write( "                         w = -1\r\n" );
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document.write( "Substitute x = 1, y = -2 and w = -1 in (3)\r\n" );
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document.write( " -(-1) + 2(1) + 5(-2) +  z = -4\r\n" );
document.write( "            1 + 2 - 10 + z = -4\r\n" );
document.write( "                    -7 + z = -4\r\n" );
document.write( "                         z = 3\r\n" );
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document.write( "(w,x,y,z) = (-1,1,-2,3)\r\n" );
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document.write( "Edwin
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