document.write( "Question 892753: use cramers rule to solve the following\r
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Algebra.Com's Answer #540773 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
use cramers rule to solve the following\r
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document.write( "Cramer's rule can only be used for independent systems, i.e.,\r\n" );
document.write( "systems that have only 1 unique solution. \r\n" );
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document.write( "If the system is inconsistent, there will be no solution,\r\n" );
document.write( "If the system is dependent there will be infinitely many\r\n" );
document.write( "solution.\r\n" );
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document.write( "We begin by finding the determinant \"D\". It consists \r\n" );
document.write( "of just the three columns of x, y, and z coefficients,\r\n" );
document.write( "in that order, but does not contain the constants, the\r\n" );
document.write( "three numbers to the right of the equal signs.\r\n" );
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document.write( "\"D=abs%28matrix%283%2C3%2C1%2C-1%2C1%2C5%2C2%2C-3%2C4%2C3%2C-4%29%29\".\r\n" );
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document.write( "It has value \"D=0\".\r\n" );
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document.write( "Therefore we cannot solve this system by Cramer's rule!\r\n" );
document.write( "However it may be a dependent system and have infinitely \r\n" );
document.write( "many solutions. \r\n" );
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document.write( "Now this is probably all your teacher wants you to do at\r\n" );
document.write( "this point.  However, later you will have to find the\r\n" );
document.write( "general solution for a dependent system, and determine if\r\n" );
document.write( "it is dependent or inconsistent.  So you should\r\n" );
document.write( "print out the rest of this and save it for future\r\n" );
document.write( "reference.  You may not study this until next year.\r\n" );
document.write( "But you can easily understand it now.\r\n" );
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document.write( "We can find the infinite set of solutions if the\r\n" );
document.write( "system turns out to be independent, or else determine that it\r\n" );
document.write( "is inconsistent.\r\n" );
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document.write( "We will create a blank space where the \r\n" );
document.write( "5x is.  Multiply the top equation by -5\r\n" );
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document.write( "Add the top equation to the middle equation\r\n" );
document.write( "and the -5x cancels the 5x, leaving a blank\r\n" );
document.write( "space where the 5x was.\r\n" );
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document.write( " \r\n" );
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document.write( "Restore the top equation as it was by dividing\r\n" );
document.write( "it by -5\r\n" );
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document.write( "Next we will create a blank space where the \r\n" );
document.write( "4x is.  Multiply the top equation by -4\r\n" );
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document.write( "Add the top equation to the middle equation\r\n" );
document.write( "and the -4x cancels the 4x, leaving a blank\r\n" );
document.write( "space where the 4x was.\r\n" );
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document.write( "Restore the top equation as it was by dividing\r\n" );
document.write( "it by -4\r\n" );
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document.write( "Now we see that the bottom equation turned out\r\n" );
document.write( "to be identical to the middle equation.\r\n" );
document.write( "So we can eliminate the bottom equation entirely\r\n" );
document.write( "by subtracting it from the middle equation, and \r\n" );
document.write( "just have:\r\n" );
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document.write( "The fact that this was possible proves that it\r\n" );
document.write( "is a dependent system, not an inconsistent\r\n" );
document.write( "system.  So there are infinitely many solutions.\r\n" );
document.write( "So let's find the general solution and some\r\n" );
document.write( "sample solutions.\r\n" );
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document.write( "So we solve the bottom equation for y:\r\n" );
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document.write( "\"7y+-+8z+=+-18\"\r\n" );
document.write( "\"7y+=+-18+%2B+8z\"\r\n" );
document.write( "\"y+=+%28-18%29%2F7+%2B+expr%288%2F7%29z\"\r\n" );
document.write( "\"y+=+-18%2F7%2Bexpr%288%2F7%29z\"\r\n" );
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document.write( "Substitute in the top equation\r\n" );
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document.write( "\"x-y%2Bz=4\"\r\n" );
document.write( "\"x-%28-18%2F7%2Bexpr%288%2F7%29z%29%2Bz=4\"\r\n" );
document.write( "\"x%2B18%2F7-expr%288%2F7%29z%2Bz=4\"\r\n" );
document.write( "Multiply through by 7 to clear the fractions:\r\n" );
document.write( "\"7x%2B18-8z%2B7z=28\"\r\n" );
document.write( "\"7x%2B18-z=28\"\r\n" );
document.write( "\"7x=10%2Bz\"\r\n" );
document.write( "\"x=10%2F7%2Bexpr%281%2F7%29z\"\r\n" );
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document.write( "So the general solution is\r\n" );
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document.write( "(x,y,z) = \r\n" );
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document.write( "Som examples of solutions:\r\n" );
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document.write( "If we choose 4 for z, we get\r\n" );
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document.write( "(x,y,z) = \r\n" );
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document.write( "(x,y,z) = \r\n" );
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document.write( "(x,y,z) = \"%28matrix%281%2C5%2C%0D%0A%0D%0A14%2F7%2C%22%2C%22%2C14%2F7%2C%22%2C%22%2C4%29%29\"\r\n" );
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document.write( "(x,y,z) = (2,2,4)\r\n" );
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document.write( "So that's a solution.\r\n" );
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document.write( "If we choose 11` for z, we get\r\n" );
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document.write( "(x,y,z) = \r\n" );
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document.write( "(x,y,z) = \r\n" );
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document.write( "(x,y,z) = \"%28matrix%281%2C5%2C%0D%0A%0D%0A21%2F7%2C%22%2C%22%2C70%2F7%2C%22%2C%22%2C11%29%29\"\r\n" );
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document.write( "(x,y,z) = (3,10,11)\r\n" );
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document.write( "So that's another solution.\r\n" );
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document.write( "If we choose -3 for z, we get\r\n" );
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document.write( "(x,y,z) = \r\n" );
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document.write( "(x,y,z) = \r\n" );
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document.write( "(x,y,z) = \"%28matrix%281%2C5%2C%0D%0A%0D%0A7%2F7%2C%22%2C%22%2C-42%2F7%2C%22%2C%22%2C-3%29%29\"\r\n" );
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document.write( "(x,y,z) = (1,-6,-3)\r\n" );
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document.write( "So that's also a solution.\r\n" );
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document.write( "We can get infinitely many solutions because it is\r\n" );
document.write( "an dependent system.  You will find that each\r\n" );
document.write( "of these three solutions we just found are solutions\r\n" );
document.write( "to the original system.\r\n" );
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document.write( "There are infinitely many solutions, and we have just\r\n" );
document.write( "found the general solution and 3 examples of solutions.\r\n" );
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document.write( "Edwin
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