document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=144078>Question 144078</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I'm Michele and I could really use someones help on my problem. I will be greatful and very thankful for who ever will help me. Thank You for your help.\r<BR>\n" );
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document.write( "1.Find k if the following system of equations has a unique solution<BR>\n" );
document.write( "2x + (k - 1)y = 6<BR>\n" );
document.write( "3x + (2k + 1)y = 9\r<BR>\n" );
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document.write( "2.Find k if the following system of equations has infinite solutions<BR>\n" );
document.write( "kx + 3y = k - 3<BR>\n" );
document.write( "12x + ky = k\r<BR>\n" );
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document.write( "3.Find k if the following system of equations has no solution<BR>\n" );
document.write( "3x + 2y = 6<BR>\n" );
document.write( "kx + (k - 1)y = 9\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #104911 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=104911\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1.Find k if the following system of equations has a unique solution<BR>\n" );
document.write( "2x + (k - 1)y = 6<BR>\n" );
document.write( "3x + (2k + 1)y = 9 <BR>\n" );
document.write( "------------------<BR>\n" );
document.write( "Multiply thru 1st by 3 ; Multiply thru 2nd by 2 to get:<BR>\n" );
document.write( "6x + (3k-3)y = 18<BR>\n" );
document.write( "6x + (4k+2)y = 18<BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "Subtract 1st from 2nd to get:<BR>\n" );
document.write( "[4k+2-(3k-3)]y = 0<BR>\n" );
document.write( "[k+5] y = 0<BR>\n" );
document.write( "Comment: y is a variable so may not always be zero<BR>\n" );
document.write( "k+5 = 0 when k=-5 <BR>\n" );
document.write( "As long as k is not -5, y will be zero and x will be 3<BR>\n" );
document.write( "So the system will have a unique solution as long as k is not equal to -5.<BR>\n" );
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document.write( "2.Find k if the following system of equations has infinite solutions<BR>\n" );
document.write( "kx + 3y = k - 3<BR>\n" );
document.write( "12x + ky = k<BR>\n" );
document.write( "-------<BR>\n" );
document.write( "The system has infinite solutions if the slopes and the intercepts are the same<BR>\n" );
document.write( "Put them in slope-intercept form:<BR>\n" );
document.write( "y = (-k/3)x + (k-3)/3  <BR>\n" );
document.write( "y = (-12/k)x + 1<BR>\n" );
document.write( "----------<BR>\n" );
document.write( "Equating slopes: -k/3 = -12/k ; k = 6<BR>\n" );
document.write( "Equating intercepts: (k-3)/3 = 1  ; k = 6<BR>\n" );
document.write( "Conclusion: infinitely many solutions if k = 6<BR>\n" );
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document.write( "Find k if the following system of equations has no solution<BR>\n" );
document.write( "3x + 2y = 6<BR>\n" );
document.write( "kx + (k - 1)y = 9 <BR>\n" );
document.write( "--------<BR>\n" );
document.write( "No solutions if slopes are the same and intercepts are not.<BR>\n" );
document.write( "y = (-3/2)x + 3<BR>\n" );
document.write( "y = (-k/(k-1)x + 9/(k-1)<BR>\n" );
document.write( "------------------------<BR>\n" );
document.write( "Equating slopes: -3/2 = -k/(k-1) ; -2k = -3k+3 ; k = 3<BR>\n" );
document.write( "Equatiing intercepts: 3 = 9/(k-1) ; k-1 = 3 ; k = 4<BR>\n" );
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document.write( "Conclusion: No solutions if k = 3<BR>\n" );
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document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
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