document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=126940>Question 126940</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0131752049/algebrahomeworkh>Thinking Mathematically</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Given the system of equations 2x-4y=10 and 26x-52y=130<BR>\n" );
document.write( "a.) How many solutions does the system have?<BR>\n" );
document.write( "b.) Find the solution(s). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #92976 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=92976\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Start with the given system of equations:\r<BR>\n" );
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document.write( "Now in order to solve this system by using elimination/addition, we need to solve (or isolate) one variable. I'm going to solve for y.\r<BR>\n" );
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document.write( "In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we would have to eliminate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or vice versa).\r<BR>\n" );
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document.write( "So lets eliminate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In order to do that, we need to have both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.\r<BR>\n" );
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document.write( "So to make the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficients equal in magnitude but opposite in sign, we need to multiply both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficients by some number to get them to an common number. So if we wanted to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=26 ALIGN=MIDDLE  ALT=\"26\"   DISABLED_style= \"max-width:90%; height:auto;\" > to some equal number, we could try to get them to the LCM.\r<BR>\n" );
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document.write( "Since the LCM of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=26 ALIGN=MIDDLE  ALT=\"26\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=26 ALIGN=MIDDLE  ALT=\"26\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we need to multiply both sides of the top equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > and multiply both sides of the bottom equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%282x-4y%29=13%2810%29 ALIGN=MIDDLE  ALT=\"13%282x-4y%29=13%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Multiply the top equation (both sides) by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2826x-52y%29=-1%28130%29 ALIGN=MIDDLE  ALT=\"-1%2826x-52y%29=-1%28130%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Multiply the bottom equation (both sides) by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Distribute and multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=26x-52y=130 ALIGN=MIDDLE  ALT=\"26x-52y=130\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-26x%2B52y=-130 ALIGN=MIDDLE  ALT=\"-26x%2B52y=-130\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now add the equations together. In order to add 2 equations, group like terms and combine them\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2826x-26x%29%2B%28-52y%2B52y%29=130-130 ALIGN=MIDDLE  ALT=\"%2826x-26x%29%2B%28-52y%2B52y%29=130-130\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Combine like terms and simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%2826x-26x%29%2B0y=0 ALIGN=MIDDLE  ALT=\"cross%2826x-26x%29%2B0y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Notice how the x terms cancel out\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=0 ALIGN=MIDDLE  ALT=\"0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "Since this equation is <b>always</b> true regardless of what x or y is, we have an infinte number of solutions. So this system is dependent.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "a.) How many solutions does the system have?\r<BR>\n" );
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document.write( "The system has an infinite number of solutions.\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "The solutions are simply points that lie on either line (since the two lines are identical)\r<BR>\n" );
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document.write( " </SPAN>\n" );
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