document.write( "Question 1085977: Find the value of k the system has infinite number of solution x+(k+1)y=5;(k+1)x+9y=8k-1
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #700089 by ikleyn(52784)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "
\r\n" );
document.write( "1.  One way to solve it is using determinant.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2.  The other way is  THIS:\r\n" );
document.write( "\r\n" );
document.write( "    the system has INFINITELY MANY solutions if and only if the two equations are equivalent.\r\n" );
document.write( "\r\n" );
document.write( "    In other words, if and only if\r\n" );
document.write( "\r\n" );
document.write( "    \"1%2F%28k%2B1%29\" = \"%28k%2B1%29%2F9\" = \"5%2F%288k-1%29\".    (1)\r\n" );
document.write( "\r\n" );
document.write( "    (the coefficients and the right side terms are proportional with the same proportionality coefficient).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "3.   From  \"1%2F%28k%2B1%29\" = \"%28k%2B1%29%2F9\"  you have  \"%28k%2B1%29%5E2\" = 9  ====>  k+1 = +/-3  ====>  a) k = -1 + 3 = 2,  and  b) k = -1 - 3 = -4.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     Now check k = 2 for this proportion: \"%28k%2B1%29%2F9\" = \"5%2F%288k-1%29\".\r\n" );
document.write( "\r\n" );
document.write( "     You have \"%282%2B1%29%2F9\" = \"1%2F3\" (left side)  and  \"5%2F%288%2A2-1%29\" = \"5%2F15\" = \"1%2F3\"  (right side).\r\n" );
document.write( "\r\n" );
document.write( "     So, the value k = 2 satisfies (1).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     Now check k = -4 for this proportion: \"%28k%2B1%29%2F9\" = \"5%2F%288k-1%29\".\r\n" );
document.write( "\r\n" );
document.write( "     You have \"%28-4%2B1%29%2F9\" = \"-1%2F3\" (left side)  and  \"5%2F%288%2A%28-4%29-1%29\" = \"5%2F%28-33%29\"  (right side).\r\n" );
document.write( "\r\n" );
document.write( "     So, the value k = -4 does not satisfy (1).\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Answer. The value of \"k\" under the question is k = 2.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "See the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Geometric interpretation of the linear system of two equations in two unknowns \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Systems of two linear equations in two unknowns\".\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );