document.write( "Question 1136571: (a) Consider the following system of linear equations:
\n" ); document.write( "x+ 5y+ z= 0
\n" ); document.write( " x+ 6y- z= 0
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\n" ); document.write( " Find values of a, b, and c such that the above system of linear equations has: (i) exactly one solution;
\n" ); document.write( "(ii) an infinite number of solution;
\n" ); document.write( "(iii) no solution
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Algebra.Com's Answer #754638 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
(a) Consider the following system of linear equations:
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\n" ); document.write( "2x +ay +bz= c
\n" ); document.write( "Find values of a, b, and c such that the above system of linear equations has:
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document.write( "\"system%28x%2B5y%2Bz=0%2C+x%2B6y-z=0%2C+2x%2Bay%2Bbz=c%29\"\r\n" );
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document.write( "Add the first and second equations:\r\n" );
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document.write( "\"2x%2B11y+=+0\"\r\n" );
document.write( "\"2x=-11y\"\r\n" );
document.write( "\"x=-11y%2F2\"\r\n" );
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document.write( "Subtract the first equation from the second:\r\n" );
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document.write( "\"y-2z=0\"\r\n" );
document.write( "\"-2z=-y\"\r\n" );
document.write( "\"z=y%2F2\"\r\n" );
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document.write( "So from the first two equations, if there are any solutions, they can only be\r\n" );
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document.write( "Substitute in the 3rd given equation: \r\n" );
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document.write( "\"2x%2Bay%2Bbz=c\"\r\n" );
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document.write( "\"2%28-11y%2F2%29%2Bay%2Bb%28y%2F2%29=c\"\r\n" );
document.write( "\"-11y%2Bay%2Bby%2F2=c\"\r\n" );
document.write( "\"-22y%2B2ay%2Bby=2c\"\r\n" );
document.write( "\"%28-22%2B2a%2Bb%29y=2c\"\r\n" );
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document.write( "Let's answer (ii) first.\r\n" );
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(ii) an infinite number of solution;
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document.write( "If -22+2a+b=0 and 2c=0 then this equation becomes\r\n" );
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document.write( "\"0y=0\", which has infinitely many solutions for y\r\n" );
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document.write( "That is when 2a+b=22 and c=0, for instance when a=7, b=8, c=0,\r\n" );
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document.write( "there are infinitely many solutions.\r\n" );
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document.write( "-------------------------------\r\n" );
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document.write( "If we solve for y\r\n" );
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document.write( " \"y=2c%2F%28-22%2B2a%2Bb%29\" \r\n" );
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document.write( "Substituting in\r\n" );
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document.write( "Let's answer (iii)\r\n" );
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(iii) no solution
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document.write( "That's when y is undefined.\r\n" );
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document.write( "So that is when the denominator -22+2a+b equals 0 and the numerator is\r\n" );
document.write( "not equal to 0, for instance when a=7, b=8, c=1, there are no solutions.\r\n" );
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document.write( "Finally we answer (i)\r\n" );
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(i) exactly one solution;
 \r\n" );
document.write( "There will be only 1 solution when -22+2a+b is not 0\r\n" );
document.write( "For instance when a=4 and b=5 and c=1.\r\n" );
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document.write( "Edwin
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