document.write( "Question 174787: For the system of equations
\n" ); document.write( " $\"system%28ax+%2B+by+=+c%2C%0D%0Adx+%2B+ey+=+f%29\"$
\n" ); document.write( "Find the conditions on a,b,c,d,e anf f, where b and e are nonzero, such that the equations have\r
\n" ); document.write( "\n" ); document.write( "i. Infinitely many solutions
\n" ); document.write( "ii. No solution
\n" ); document.write( "iii. A unique solution \n" ); document.write( "

Algebra.Com's Answer #129866 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!
For the system of equations
\n" ); document.write( " $\"system%28ax+%2B+by+=+c%2C%0D%0Adx+%2B+ey+=+f%29\"$
\n" ); document.write( "Find the conditions on a,b,c,d,e and f, where
\n" ); document.write( "b and e are nonzero, such that the equations have
\n" ); document.write( "

\r\n" );
document.write( "There are more than one way to do this, and\r\n" );
document.write( "the method used depends on what course you\r\n" );
document.write( "are taking.  It could be done with determinants,\r\n" );
document.write( "matrices, or neither.  And there are different \r\n" );
document.write( "ways in each of those.  I will arbitrarily do it \r\n" );
document.write( "one way with neither.  If you're studying another \r\n" );
document.write( "method then post again telling us what method \r\n" );
document.write( "you are studying. \r\n" );
document.write( "\r\n" );
document.write( "-----\r\n" );
document.write( "\r\n" );
document.write( "Equations like these\r\n" );
document.write( "\r\n" );
document.write( "i.   and  \r\n" );
document.write( "\r\n" );
document.write( "have infinitely many solutions\r\n" );
document.write( "since any number may be \r\n" );
document.write( "substituted for x or y and the\r\n" );
document.write( "the equation will \r\n" );
document.write( "always become ,\r\n" );
document.write( "which will always be true.\r\n" );
document.write( "\r\n" );
document.write( "Equations like these\r\n" );
document.write( "\r\n" );
document.write( "ii.  and \r\n" );
document.write( "\r\n" );
document.write( "where neither r nor s is 0\r\n" );
document.write( "\r\n" );
document.write( "cannot have any solutions,\r\n" );
document.write( "because the left side will\r\n" );
document.write( "always be 0 and the right\r\n" );
document.write( "side will be something other\r\n" );
document.write( "than 0, which will never be \r\n" );
document.write( "true.\r\n" );
document.write( "\r\n" );
document.write( "iii.  Equations like these\r\n" );
document.write( "\r\n" );
document.write( "ii.  and \r\n" );
document.write( "\r\n" );
document.write( "where neither p nor q is 0  will\r\n" );
document.write( "have exactly one unique solution, \r\n" );
document.write( "for since neither the coefficient\r\n" );
document.write( "of x and y is  0, then we can\r\n" );
document.write( "divide both sides by that non-zero\r\n" );
document.write( "coefficient and get one and only\r\n" );
document.write( "one solution.\r\n" );
document.write( "\r\n" );
document.write( "----------------------\r\n" );
document.write( "\r\n" );
document.write( "Let's start to solve the system by\r\n" );
document.write( "elimination:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Multiply the first equation through by \r\n" );
document.write( "and multiply the second equation through by \r\n" );
document.write( "to eliminate the x's\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Adding corresponding terms, the x-terms cancel\r\n" );
document.write( "out and we have:\r\n" );
document.write( "\r\n" );
document.write( "    \r\n" );
document.write( "\r\n" );
document.write( "Factor out y on the left:\r\n" );
document.write( "\r\n" );
document.write( "    \r\n" );
document.write( "\r\n" );
document.write( "------\r\n" );
document.write( "\r\n" );
document.write( "Now we eliminate the y-terms.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Multiply the first equation through by \r\n" );
document.write( "and multiply the second equation through by \r\n" );
document.write( "to eliminate the x's\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Adding corresponding terms, the y-terms cancel\r\n" );
document.write( "out and we have:\r\n" );
document.write( "\r\n" );
document.write( "    \r\n" );
document.write( "\r\n" );
document.write( "Factor out x on the left:\r\n" );
document.write( "\r\n" );
document.write( "    \r\n" );
document.write( "\r\n" );
document.write( "So we have the system:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "

\n" ); document.write( "i. Infinitely many solutions
\n" ); document.write( "

\r\n" );
document.write( "\r\n" );
document.write( "Here we need to have a case of\r\n" );
document.write( "\r\n" );
document.write( ", so we set the \r\n" );
document.write( "coefficient of x and y on the left, which are\r\n" );
document.write( "both , equal to 0:\r\n" );
document.write( "\r\n" );
document.write( " or \r\n" );
document.write( "\r\n" );
document.write( "We also set each right side equal to 0.\r\n" );
document.write( "Setting the first right side = 0,\r\n" );
document.write( "\r\n" );
document.write( " or  \r\n" );
document.write( "\r\n" );
document.write( "Setting the second right side = 0,\r\n" );
document.write( "\r\n" );
document.write( " or \r\n" );
document.write( "\r\n" );
document.write( "Putting these results toghether, these fractions\r\n" );
document.write( "are all equal, so the answer to (i) is\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "

\n" ); document.write( "ii. no solution
\n" ); document.write( "

\r\n" );
document.write( " and \r\n" );
document.write( "\r\n" );
document.write( "where neither p nor q is 0, so\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This system is like (i) in that we have\r\n" );
document.write( "\r\n" );
document.write( " or \r\n" );
document.write( "\r\n" );
document.write( "However it is unlike (i) in that we must \r\n" );
document.write( "require that the right sides NOT equal 0,\r\n" );
document.write( "\r\n" );
document.write( "Setting the first right side NOT equal 0,\r\n" );
document.write( "\r\n" );
document.write( " or  \r\n" );
document.write( "\r\n" );
document.write( "Setting the second right side NOT equal 0,\r\n" );
document.write( "\r\n" );
document.write( " or \r\n" );
document.write( "\r\n" );
document.write( "Putting these results together, the answer \r\n" );
document.write( "to (ii) is\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "---\r\n" );
document.write( "

\n" ); document.write( "iii. A unique solution
\n" ); document.write( "

\r\n" );
document.write( "There will be exactly one\r\n" );
document.write( "unique solution if the coefficient\r\n" );
document.write( "of x and y is not 0, for then we can\r\n" );
document.write( "divide both sides by that non-zero\r\n" );
document.write( "coefficient and get one and only\r\n" );
document.write( "one solution.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So we require that , so that\r\n" );
document.write( "we can divide both sides of both equations\r\n" );
document.write( "by  and get just one value for x\r\n" );
document.write( "and just one value for y.\r\n" );
document.write( "\r\n" );
document.write( "So we just have to require that\r\n" );
document.write( "\r\n" );
document.write( " or \r\n" );
document.write( "\r\n" );
document.write( "So the answer to (iii) is\r\n" );
document.write( "\r\n" );
document.write( " and there are no restrictions on\r\n" );
document.write( " or .\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "

\n" );