document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=133357>Question 133357</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the points of intersection of the ellipse 16x^2 + 25y^2 = 400 and the line 5y + 4x = 20\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 #97543 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=97543\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In order to solve this system\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+25y%5E2+=+400 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+25y%5E2+=+400\"   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=5y+%2B+4x+=+20 ALIGN=MIDDLE  ALT=\"5y+%2B+4x+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We need to isolate one variable and substitute that expression to eliminate one variable.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y+%2B+4x+=+20 ALIGN=MIDDLE  ALT=\"5y+%2B+4x+=+20\"   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=5y+=+20-4x ALIGN=MIDDLE  ALT=\"5y+=+20-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 4x from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%2820-4x%29%2F5 ALIGN=MIDDLE  ALT=\"y+=+%2820-4x%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 5 to isolate y\r<BR>\n" );
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document.write( "---------------\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+25y%5E2+=+400 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+25y%5E2+=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Go back to the first equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+25%28%2820-4x%29%2F5%29%5E2+=+400 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+25%28%2820-4x%29%2F5%29%5E2+=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%2820-4x%29%2F5 ALIGN=MIDDLE  ALT=\"y+=+%2820-4x%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice how we eliminated the \"y\" terms and now have one equation with one unknown.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+25%28%2820-4x%29%5E2%2F5%5E2%29+=+400 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+25%28%2820-4x%29%5E2%2F5%5E2%29+=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute the outer exponent\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+25%28%2820-4x%29%5E2%2F25%29+=+400 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+25%28%2820-4x%29%5E2%2F25%29+=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E2 ALIGN=MIDDLE  ALT=\"5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 25\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+cross%2825%29%28%2820-4x%29%5E2%2Fcross%2825%29%29+=+400 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+cross%2825%29%28%2820-4x%29%5E2%2Fcross%2825%29%29+=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+%2820-4x%29%5E2+=+400 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+%2820-4x%29%5E2+=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+400-160x%2B16x%5E2+=+400 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+400-160x%2B16x%5E2+=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Foil\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2+%2B+400-160x%2B16x%5E2+-+400=0 ALIGN=MIDDLE  ALT=\"16x%5E2+%2B+400-160x%2B16x%5E2+-+400=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 400 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32x%5E2+-160x=0 ALIGN=MIDDLE  ALT=\"32x%5E2+-160x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32x%28x+-5%29=0 ALIGN=MIDDLE  ALT=\"32x%28x+-5%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32x ALIGN=MIDDLE  ALT=\"32x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now set each factor equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32x=0 ALIGN=MIDDLE  ALT=\"32x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-5=0 ALIGN=MIDDLE  ALT=\"x+-5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now solve for x for each factor:\r<BR>\n" );
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document.write( "So our solutions are \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "So let's find y when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   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=5y+%2B+4x+=+20 ALIGN=MIDDLE  ALT=\"5y+%2B+4x+=+20\"   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=5y+%2B+4%280%29+=+20 ALIGN=MIDDLE  ALT=\"5y+%2B+4%280%29+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   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=5y+=+20 ALIGN=MIDDLE  ALT=\"5y+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply and simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+4 ALIGN=MIDDLE  ALT=\"y+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 5 to isolate x\r<BR>\n" );
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document.write( "So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4 ALIGN=MIDDLE  ALT=\"y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So one intersection point is (0,4)\r<BR>\n" );
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document.write( "So let's find y when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   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=5y+%2B+4x+=+20 ALIGN=MIDDLE  ALT=\"5y+%2B+4x+=+20\"   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=5y+%2B+4%285%29+=+20 ALIGN=MIDDLE  ALT=\"5y+%2B+4%285%29+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   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=5y+%2B20=+20 ALIGN=MIDDLE  ALT=\"5y+%2B20=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply and simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y+=0 ALIGN=MIDDLE  ALT=\"5y+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 20 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+0 ALIGN=MIDDLE  ALT=\"y+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 5 to isolate x\r<BR>\n" );
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document.write( "So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So one intersection point is (5,0)\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So our solutions are  (0,4) and (5,0)\r<BR>\n" );
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document.write( "Notice if we graph, we can visually verify our answer\r<BR>\n" );
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