document.write( "Question 563136: Hi --\r
\n" ); document.write( "\n" ); document.write( "I'm a beginning student in Linear Algebra, and am not quite sure how to answer the following question, which is taken from my textbook:\r
\n" ); document.write( "\n" ); document.write( "\"Suppose the system below is consistent for all possible values of f and g. What can you say about the coefficients c and d. Justify your answer.\r
\n" ); document.write( "\n" ); document.write( "2x + 4y = f
\n" ); document.write( "cx + dy = g.\"\r
\n" ); document.write( "\n" ); document.write( "Not quite sure what is meant by the question \"what can you say.\" My only thought so far is that If the system is consistent, that means x and y have the same value for both equations:\r
\n" ); document.write( "\n" ); document.write( "x(2 + c) + y(4 + d) = f + g, which, if c = f+g, fits into the form ax + by = c. But that's about it. \r
\n" ); document.write( "\n" ); document.write( "Thanks in advance!\r
\n" ); document.write( "\n" ); document.write( "Cheers,
\n" ); document.write( "Dave Yrueta \n" ); document.write( "
Algebra.Com's Answer #364768 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "No, they are consistent if they have a solution.\r\n" );
document.write( "So we find what their solution must be by elimination:\r\n" );
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document.write( "2x + 4y = f\r\n" );
document.write( "cx + dy = g\r\n" );
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document.write( "Multiply the first equation through by -d and the \r\n" );
document.write( "second equation through by 4\r\n" );
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document.write( "-2dx - 4dy = -df\r\n" );
document.write( " 4cx + 4dy =  4g\r\n" );
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document.write( "(4c-2d)x = 4g-df\r\n" );
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document.write( "       x = \"%284g-df%29%2F%284c-2d%29\"\r\n" );
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document.write( "Start over:\r\n" );
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document.write( "2x + 4y = f\r\n" );
document.write( "cx + dy = g\r\n" );
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document.write( "Multiply the first equation through by c and the second equation through\r\n" );
document.write( "by -2\r\n" );
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document.write( " 2cx + 4cy =  cf\r\n" );
document.write( "-2cx - 2cy = -2g\r\n" );
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document.write( "(4c-2d)y = cf-26\r\n" );
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document.write( "       y = \"%28cf-2g%29%2F%284c-2d%29\"\r\n" );
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document.write( "Notice that for both x and y, the denominators\r\n" );
document.write( "are the same, 4c-2d\r\n" );
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document.write( "In order for those expresions for x and y to be\r\n" );
document.write( "defined solution, their denominators must not be\r\n" );
document.write( "zero, because division by 0 is not defined. But\r\n" );
document.write( "division by any other number is defined.  Therefore \r\n" );
document.write( "the only requirement is that their denominator must \r\n" );
document.write( "not be equal to 0, so\r\n" );
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document.write( "       4c-2d ≠ 0\r\n" );
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document.write( "divide through by 2\r\n" );
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document.write( "        2c-d ≠ 0\r\n" );
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document.write( "          2c ≠ d\r\n" );
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document.write( "           d ≠ 2c\r\n" );
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document.write( "As long as d is not equal to 2c, there will be a solution for\r\n" );
document.write( "any values of f and g.   \r\n" );
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document.write( "The answer to: \"What can you say about the coefficients c and d?\" is\r\n" );
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document.write( "d is not equal to 2c.\r\n" );
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document.write( "[If you have studied Cramer's rule, you can just say the determinant of\r\n" );
document.write( "coefficients D must not equal 0.  The above was done assuming you have not\r\n" );
document.write( "studied or were not allowed to use Cramer's rule.]\r\n" );
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document.write( "Edwin
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