document.write( "Question 549768: Tell how to determine whether the system has a unique solution, and solve it with the Cramer method:\r
\n" ); document.write( "\n" ); document.write( "ax + by = c
\n" ); document.write( "dx + ey = f\r
\n" ); document.write( "\n" ); document.write( "The answer is: the system has a unique solution if:
\n" ); document.write( " delta = ae - bd not equal to 0
\n" ); document.write( "and the solution is:
\n" ); document.write( "x = (ce - bf) / (ae - bd)
\n" ); document.write( "y = (af - cd) / (ae - bd)\r
\n" ); document.write( "\n" ); document.write( "How do we explain that?\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "

Algebra.Com's Answer #358110 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\r\n" );
document.write( "ax + by = c\r\n" );
document.write( "dx + ey = f\r\n" );
document.write( "\r\n" );
document.write( "By Cramer's rule:\r\n" );
document.write( "\r\n" );
document.write( "Delta = \"abs%28matrix%282%2C2%2Ca%2Cb%2Cd%2Ce%29%29\" = ae - bd\r\n" );
document.write( "\r\n" );
document.write( "Dx = \"abs%28matrix%282%2C2%2Cc%2Cb%2Cf%2Ce%29%29\" = ce - bf \r\n" );
document.write( "\r\n" );
document.write( "Dy = \"abs%28matrix%282%2C2%2Ca%2Cc%2Cd%2Cf%29%29\" = af - cd\r\n" );
document.write( "\r\n" );
document.write( "x = \"D%5Bx%5D%2F%28Delta%29\" = \"abs%28matrix%282%2C2%2Cc%2Cb%2Cf%2Ce%29%29%2F%28ae-bd%29\" = \"%28ce-bf%29%2F%28ae-bd%29\"\r\n" );
document.write( "\r\n" );
document.write( "y = \"D%5By%5D%2F%28Delta%29\" = \"abs%28matrix%282%2C2%2Ca%2Cc%2Cd%2Cf%29%29%2F%28ae-bd%29\" = \"%28af-cd%29%2F%28ae-bd%29\"\r\n" );
document.write( "\r\n" );
document.write( "Denominators cannot be 0, so Delta = \"abs%28matrix%282%2C2%2Ca%2Cb%2Cd%2Ce%29%29\" = ae - bd ≠ 0.\r\n" );
document.write( "and both \"%28ce-fe%29%2F%28ae-bd%29\" and \"%28af-cd%29%2F%28ae-bd%29\" have unique\r\n" );
document.write( "values when that denominator is not zero.\r\n" );
document.write( "\r\n" );
document.write( "------------------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "In case you need to explain why Cramer's rule works:\r\n" );
document.write( " \r\n" );
document.write( "To eliminate y, multiply the first equation by e and the\r\n" );
document.write( "second equation by -b, and add the equations vertically\r\n" );
document.write( "term by term:\r\n" );
document.write( "\r\n" );
document.write( "     aex + bey =  ce\r\n" );
document.write( "    -bdx - bey = -bf\r\n" );
document.write( "-----------------------\r\n" );
document.write( " aex-bdx       =  ce-bf  \r\n" );
document.write( "(ae-bd)x       =  ce-bf\r\n" );
document.write( "       x = \"%28ce-bf%29%2F%28ae-bd%29\"\r\n" );
document.write( "\r\n" );
document.write( "To eliminate x, multiply the first equation by -d and the\r\n" );
document.write( "second equation by a, and add the equations vertically\r\n" );
document.write( "term by term:\r\n" );
document.write( "\r\n" );
document.write( "  -adx   - bdy = -cd\r\n" );
document.write( "   adx   + aey =  af\r\n" );
document.write( "-----------------------\r\n" );
document.write( "       aey-bdy =  af-cd  \r\n" );
document.write( "      (ae-bd)y =  af-cd\r\n" );
document.write( "             y = \"%28af-cd%29%2F%28ae-bd%29\"\r\n" );
document.write( "\r\n" );
document.write( "a and y are the same using the elimination method as they are \r\n" );
document.write( "using Cramer's rule.\r\n" );
document.write( "  \r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );