document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252412>Question 252412</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have been sitting here for hours trying to figure the following problems.  I am working on Non-Linear Systems and have to solve the following systems.  No matter how I work it I can't seem to come to a solution.\r<BR>\n" );
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document.write( "1)  {7x-8y=24<BR>\n" );
document.write( "     {xy^2=1\r<BR>\n" );
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document.write( "2)  {(x+1)^2 - (y-1)^2 = 20<BR>\n" );
document.write( "    {x^2 - (y+2)^2 - 24\r<BR>\n" );
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document.write( "Thank you - Lori </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184310 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=184310\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Even though these are non-linear equations, we can still use substitution to solve them. I'll do the first one to get you started. If that doesn't help either repost or ask me.\r<BR>\n" );
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document.write( "# 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%5E2=1 ALIGN=MIDDLE  ALT=\"xy%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the second equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F%28y%5E2%29 ALIGN=MIDDLE  ALT=\"x=1%2F%28y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by {{y^2}}} to isolate 'x'.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x-8y=24 ALIGN=MIDDLE  ALT=\"7x-8y=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Move onto the first equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%281%2F%28y%5E2%29%29-8y=24 ALIGN=MIDDLE  ALT=\"7%281%2F%28y%5E2%29%29-8y=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F%28y%5E2%29 ALIGN=MIDDLE  ALT=\"x=1%2F%28y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-8y%5E3=24y%5E2 ALIGN=MIDDLE  ALT=\"7-8y%5E3=24y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply EVERY term by the LCD <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to clear out the fractions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8y%5E3-24y%5E2%2B7=0 ALIGN=MIDDLE  ALT=\"-8y%5E3-24y%5E2%2B7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Get every term to the left side.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E3%2B24y%5E2-7=0 ALIGN=MIDDLE  ALT=\"8y%5E3%2B24y%5E2-7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply every term by -1.\r<BR>\n" );
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document.write( "Now use the rational root theorem to find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2F2 ALIGN=MIDDLE  ALT=\"y=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a root to the polynomial equation above. In other words, if you plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2F2 ALIGN=MIDDLE  ALT=\"y=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E3%2B24y%5E2-7 ALIGN=MIDDLE  ALT=\"8y%5E3%2B24y%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you will get 0. Because of this fact, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y-1 ALIGN=MIDDLE  ALT=\"2y-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E3%2B24y%5E2-7 ALIGN=MIDDLE  ALT=\"8y%5E3%2B24y%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now use polynomial long division to find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288y%5E3%2B24y%5E2-7%29%2F%282y-1%29=4y%5E2%2B14y%2B7 ALIGN=MIDDLE  ALT=\"%288y%5E3%2B24y%5E2-7%29%2F%282y-1%29=4y%5E2%2B14y%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E3%2B24y%5E2-7=%282y-1%29%284y%5E2%2B14y%2B7%29 ALIGN=MIDDLE  ALT=\"8y%5E3%2B24y%5E2-7=%282y-1%29%284y%5E2%2B14y%2B7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This tells us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282y-1%29%284y%5E2%2B14y%2B7%29=0 ALIGN=MIDDLE  ALT=\"%282y-1%29%284y%5E2%2B14y%2B7%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y-1=0 ALIGN=MIDDLE  ALT=\"2y-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives a root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can ignore this equation. So the next step is to solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2%2B14y%2B7=0 ALIGN=MIDDLE  ALT=\"4y%5E2%2B14y%2B7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for 'y'. Use the quadratic equation to find the next two solutions of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-7%2Bsqrt%2821%29%29%2F4 ALIGN=MIDDLE  ALT=\"y=%28-7%2Bsqrt%2821%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-7-sqrt%2821%29%29%2F4 ALIGN=MIDDLE  ALT=\"y=%28-7-sqrt%2821%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the three solutions in terms of 'y' are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2F2 ALIGN=MIDDLE  ALT=\"y=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-7%2Bsqrt%2821%29%29%2F4 ALIGN=MIDDLE  ALT=\"y=%28-7%2Bsqrt%2821%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-7-sqrt%2821%29%29%2F4 ALIGN=MIDDLE  ALT=\"y=%28-7-sqrt%2821%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From here, plug each solution (in terms of 'y') into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F%28y%5E2%29 ALIGN=MIDDLE  ALT=\"x=1%2F%28y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the corresponding solution in terms of 'x'.\r<BR>\n" );
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document.write( "I skipped a bit of steps (since they're a bit long and I'm out of time for now), so feel free to ask about any step. </SPAN>\n" );
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