document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=712086>Question 712086</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the range of values which the line y=kx+3 meets the curve xy+20=5y </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #437739 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39617)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=437739\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You should be able to substitute y=kx+3 into xy+20=5y and solve for k.  That should help to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kx%5E2%2B3%281-5k%29x%2B5=0 ALIGN=MIDDLE  ALT=\"kx%5E2%2B3%281-5k%29x%2B5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Knowledge of the general solution to a quadratic equation suggests k must not be zero.\r<BR>\n" );
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document.write( "Carry out the steps you find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-3%281-5k%29-sqrt%28225k%5E2-110k%2B9%29%29%2F%282k%29 ALIGN=MIDDLE  ALT=\"x=%28-3%281-5k%29-sqrt%28225k%5E2-110k%2B9%29%29%2F%282k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-3%281-5k%29%2Bsqrt%28225k%5E2-110k%2B9%29%29%2F%282k%29 ALIGN=MIDDLE  ALT=\"x=%28-3%281-5k%29%2Bsqrt%28225k%5E2-110k%2B9%29%29%2F%282k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From that you may want to be sure to know how the discriminant should be greater than or equal to zero.  Solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=225k%5E2-110k%2B9%3E=0 ALIGN=MIDDLE  ALT=\"225k%5E2-110k%2B9%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Again use solution to quadratic equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%2811-2%2Asqrt%2810%29%29%2F45 ALIGN=MIDDLE  ALT=\"k=%2811-2%2Asqrt%2810%29%29%2F45\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%2811%2B2%2Asqrt%2810%29%29%2F45 ALIGN=MIDDLE  ALT=\"k=%2811%2B2%2Asqrt%2810%29%29%2F45\"   DISABLED_style= \"max-width:90%; height:auto;\" >  But those values should be first checked as critical points.  The k quadratic should be greater than OR EQUAL TO ZERO.  There are three intervals of k to check the discriminant expression as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=225k%5E2-110k%2B9%3E=0 ALIGN=MIDDLE  ALT=\"225k%5E2-110k%2B9%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Pushing ahead according to testing k in those intervals, the critical points in decimalized form would be close to k at 0.10389 and at 0.385.  I picked to check values 0, 0.2, and 0.4.  The results of the expression>=0 went like this:\r<BR>\n" );
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document.write( "at 0:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%3E=0 ALIGN=MIDDLE  ALT=\"9%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, yes.<BR>\n" );
document.write( "at 0.2: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%3E=0 ALIGN=MIDDLE  ALT=\"-4%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, no.<BR>\n" );
document.write( "at 0.4: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%3E=0 ALIGN=MIDDLE  ALT=\"1%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, yes.\r<BR>\n" );
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document.write( "Based on that, it seems k should be this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3C=%2811-2%2Asqrt%2810%29%29%2F45 ALIGN=MIDDLE  ALT=\"k%3C=%2811-2%2Asqrt%2810%29%29%2F45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "OR<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3E=%2811%2B2%2Asqrt%2810%29%29%2F45 ALIGN=MIDDLE  ALT=\"k%3E=%2811%2B2%2Asqrt%2810%29%29%2F45\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This was based on expecting complex values of k to cause lack of meaningful solutions to the original set of equations. </SPAN>\n" );
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