document.write( "Question 1092265: Find two numbers whose sum is 11 and whose difference between their squares is 33 \n" ); document.write( "
Algebra.Com's Answer #706865 by ikleyn(52915)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "
\r\n" );
document.write( "We are given these two equalities\r\n" );
document.write( "\r\n" );
document.write( "x   +  y  = 11,      (1)\r\n" );
document.write( "x^2 - y^2 = 33.      (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Factor (2) in this way:\r\n" );
document.write( "\r\n" );
document.write( "x^2 - y^2 = (x+y)*(x-y).  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then (2) takes the form\r\n" );
document.write( "\r\n" );
document.write( "(x+y)*(x-y) = 33.    (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now replace (x+y) in (3) by 11,  based on (1). You will get\r\n" );
document.write( "\r\n" );
document.write( "11*(x-y) = 33,    or\r\n" );
document.write( "\r\n" );
document.write( "x - y = 3.          (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus equalities (1) and (2), taken together, are equivalent to\r\n" );
document.write( "\r\n" );
document.write( "x + y = 11,         (5)\r\n" );
document.write( "x - y = 3.          (6)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now add  (5) and (6) (both sides).  You will get\r\n" );
document.write( "\r\n" );
document.write( "2x = 11+ 3 = 14.  Hence,  x = \"14%2F2\" = 7.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then from (5)  y = 11-x = 11 - 7 = 4.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Answer.  The two numbers are 7 and 4.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "--------------
\n" ); document.write( "One can solve it by another way, by substituting  y = 11-x  from  (1)  into  (2)  and solving a quadratic equation.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "I prefer my way,  since it allows to avoid quadratic equation.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---------------
\n" ); document.write( "The plot below is for illustration.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Plot y = \"sqrt%28x%5E2-33%29\" and y = 11-x\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );