document.write( "Question 1176796: If the linear system
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\n" ); document.write( "-6x+5y+6z=-5
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\n" ); document.write( "-6x+23y+hz=k\r
\n" ); document.write( "\n" ); document.write( "has infinitely many solutions, then k= and h= \n" ); document.write( "
Algebra.Com's Answer #804645 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "-6x +  5y + 6z = -5\r\n" );
document.write( " 3x +  2y + 6z = -5\r\n" );
document.write( "-6x + 23y + hz =  k\r\n" );
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document.write( "For this to have infinitely many solutions, the coefficient determinant must\r\n" );
document.write( "equal 0:\r\n" );
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document.write( "\"matrix%281%2C2%2Ccoefficient%2Cmatrix%29\"\"%22%22=%22%22\"\"abs%28matrix%283%2C3%2C-6%2C5%2C6%2C%0D%0A+3%2C2%2C6%2C-6%2C23%2Ch%29%29\"\r\n" );
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document.write( "We evaluate the determinant by copying over the first two columns on the\r\n" );
document.write( "right\r\n" );
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document.write( "\"abs%28matrix%283%2C3%2C-6%2C5%2C6%2C%0D%0A+3%2C2%2C6%2C-6%2C23%2Ch%29%29\"\"matrix%283%2C2%2C-6%2C5%2C3%2C2%2C-6%2C23%29\"\r\n" );
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document.write( "Add the products of the three upper left to lower right diagonal elements:\r\n" );
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document.write( "(-6)(2)(h)+(5)(6)(-6)+(6)(3)(23)\r\n" );
document.write( "        -12h-180+414\r\n" );
document.write( "          -12h+234\r\n" );
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document.write( "Now subtract the products of the three upper right to lower left diagonal elements:\r\n" );
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document.write( "-(6)(2)(-6)-(-6)(6)(23)-(5)(3)(h)\r\n" );
document.write( "          72+828-15h\r\n" );
document.write( "           900-15h\r\n" );
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document.write( "We combine those and get -12h+234+900-15h = -27h+1134\r\n" );
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document.write( "Set that equal to 0\r\n" );
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document.write( "-27h+1134 = 0\r\n" );
document.write( "     -27h = -1134\r\n" );
document.write( "        h = 42\r\n" );
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document.write( "Now our system becomes:\r\n" );
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document.write( "-6x +  5y +  6z = -5\r\n" );
document.write( " 3x +  2y +  6z = -5\r\n" );
document.write( "-6x + 23y + 42z =  k\r\n" );
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document.write( "Eliminate z from the first two by multiplying the\r\n" );
document.write( "1st equation by -1 and adding.  That gives:\r\n" );
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document.write( "        9x - 3y = 0\r\n" );
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document.write( "Now eliminate z from the second and third by\r\n" );
document.write( "multiplying the second by -7 and adding.  \r\n" );
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document.write( "  -21x - 14y - 42z = 35\r\n" );
document.write( "   -6x + 23y + 42z =  k\r\n" );
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document.write( "  -27x +  9y       = 35+k\r\n" );
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document.write( "Now we multiply the equation 9x - 3y = 0 through\r\n" );
document.write( "by 3 and add it to that result:\r\n" );
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document.write( "        -27x +  9y = 35+k\r\n" );
document.write( "         27x -  9y = 0\r\n" );
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document.write( "                 0 = 35+k\r\n" );
document.write( "                35 = k\r\n" );
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document.write( "So h = 42 and k = 35\r\n" );
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document.write( "That's the answer.  But eventually, you'll have to give the solution\r\n" );
document.write( "in terms of z.  When you do, the solution will be:\r\n" );
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document.write( "The solution is (x,y,z) = \"%28matrix%281%2C5%2C-5%2F9-expr%282%2F3%29z%2C+%22%2C%22%2C+-5%2F3-2z%2C%22%2C%22%2Cz%29%29\"\r\n" );
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document.write( "Edwin
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