document.write( "Question 575606: Consider the system
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Algebra.Com's Answer #369556 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "3x + ky = 0\r\n" );
document.write( "2x + 2y = 0\r\n" );
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document.write( "Solve \r\n" );
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document.write( "2x + 2y = 0 for y\r\n" );
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document.write( "2x + 2y = 0  divide through by 2\r\n" );
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document.write( "  x + y = 0\r\n" );
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document.write( "      y = -x\r\n" );
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document.write( "Substitute in \r\n" );
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document.write( "3x + ky = 0\r\n" );
document.write( "3x + k(-x) = 0\r\n" );
document.write( "3x - kx = 0\r\n" );
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document.write( "x(3 - k) = 0\r\n" );
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document.write( "Either x = 0 or else k = 3\r\n" );
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document.write( "If k ≠ 3 then x = 0, and y = -x = -(0) = 0, \r\n" );
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document.write( "and (x,y) = (0,0) is the only (unique) solution.\r\n" );
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document.write( "However if k = 3, then x can be chosen as any number,\r\n" );
document.write( "and y will equal -x, [the opposite of whatever\r\n" );
document.write( "was chosen for x].\r\n" );
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document.write( "If k = 3, then there is not just one (unique) solution, \r\n" );
document.write( "since (x,y) = (a,-a) will always be a solution to the \r\n" );
document.write( "system as long as k = 3:\r\n" );
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document.write( "3x + 3y = 0\r\n" );
document.write( "2x + 2y = 0\r\n" );
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document.write( "For instance, not only is (x,y)=(0,0)  a solution to that\r\n" );
document.write( "system, but also (1,-1), (-1,1), (-2,2), (3.1416,-3.1416),\r\n" );
document.write( "(-1000000,1000000) and infinitely many more are solutions. \r\n" );
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document.write( "However if k is any other value besides 3, there is just one\r\n" );
document.write( "(unique) solution, (x,y) = (0,0).  For example, these systems:\r\n" );
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document.write( "3x + 2y = 0   3x - 7y = 0    3x =  y = 0    3x + 999y = 0\r\n" );
document.write( "2x + 2y = 0,  3x + 2y = 0,   2x + 2y = 0,   3x +   2y = 0\r\n" );
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document.write( "all have just one (unique) solution (x,y)=(0,0), since k ≠ 3\r\n" );
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document.write( "Bottom line:  \r\n" );
document.write( "unique solution (0,0) if k ≠ 3, \r\n" );
document.write( "infinitely many solutions (a,-a) if k = 3, where a is any real number\r\n" );
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document.write( "Edwin
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