document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=121806>Question 121806</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would solve this system by addition?<BR>\n" );
document.write( "2x - 4y = 7<BR>\n" );
document.write( "4x - 2y = 9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #89425 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=89425\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+-+4y+=+7 ALIGN=MIDDLE  ALT=\"2x+-+4y+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+-+2y+=+9 ALIGN=MIDDLE  ALT=\"4x+-+2y+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You really don't want to solve this system by 'addition.'  What you want is to solve it by elimination, using addition as one of the steps of the process.\r<BR>\n" );
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document.write( "The idea of solving by elimination is to multiply (if necessary) one of the equations by some constant so that one of the variables will have a coefficient that is the additive inverse of the coefficient on the same variable in the other equation.\r<BR>\n" );
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document.write( "In this example, multiply the first equation by -2, resulting in a coefficient on the x term of -4.  -4 is the additive inverse of the 4 coefficient on x in the second equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4x+%2B+8y+=+-14 ALIGN=MIDDLE  ALT=\"-4x+%2B+8y+=+-14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x-2y=9 ALIGN=MIDDLE  ALT=\"4x-2y=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now you can add the two equations, term-by-term.  That is, you add the x terms to the x terms, the y terms to the y terms, and the constants to the constants, resulting in one equation with a zero coefficient on one of the variables, in this case, x.\r<BR>\n" );
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document.write( "The sum equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0x%2B6y=-5 ALIGN=MIDDLE  ALT=\"0x%2B6y=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Divide by 6:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%285%2F6%29 ALIGN=MIDDLE  ALT=\"y=-%285%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now that you have a value for y, you can either substitute that value into either one of the original equations and solve for x, or you could take the original two equations, multiply one of them by an appropriate constant to eliminate the y variable so that you can solve for x.  Either way works.  Watch carefully:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x-2%28-5%2F6%29=9 ALIGN=MIDDLE  ALT=\"4x-2%28-5%2F6%29=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B5%2F3=9 ALIGN=MIDDLE  ALT=\"4x%2B5%2F3=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x=%2827%2F3%29-%285%2F3%29 ALIGN=MIDDLE  ALT=\"4x=%2827%2F3%29-%285%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x=22%2F3 ALIGN=MIDDLE  ALT=\"4x=22%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=11%2F6 ALIGN=MIDDLE  ALT=\"x=11%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Or:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+-+4y+=+7 ALIGN=MIDDLE  ALT=\"2x+-+4y+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+-+2y+=+9 ALIGN=MIDDLE  ALT=\"4x+-+2y+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "2nd equation times -2:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+-+4y+=+7 ALIGN=MIDDLE  ALT=\"2x+-+4y+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8x+%2B+4y+=+-18 ALIGN=MIDDLE  ALT=\"-8x+%2B+4y+=+-18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Add:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6x=-11 ALIGN=MIDDLE  ALT=\"-6x=-11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=11%2F6 ALIGN=MIDDLE  ALT=\"x=11%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Achieving the same result.\r<BR>\n" );
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document.write( "Therefore your solution set is the ordered pair <b><big>(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2F6 ALIGN=MIDDLE  ALT=\"11%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%285%2F6%29 ALIGN=MIDDLE  ALT=\"-%285%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)</b></big>\r<BR>\n" );
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document.write( "Check your answer:\r<BR>\n" );
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document.write( "First, algebraically:\r<BR>\n" );
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document.write( "Is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+-+4y+=+7 ALIGN=MIDDLE  ALT=\"2x+-+4y+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" > true @ <b><big>(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2F6 ALIGN=MIDDLE  ALT=\"11%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%285%2F6%29 ALIGN=MIDDLE  ALT=\"-%285%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)</b></big>?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2811%2F6%29-4%28-%285%2F6%29%29=%2822%2F6%29%2B%2820%2F6%29=42%2F6=7 ALIGN=MIDDLE  ALT=\"2%2811%2F6%29-4%28-%285%2F6%29%29=%2822%2F6%29%2B%2820%2F6%29=42%2F6=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Check.\r<BR>\n" );
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document.write( "Is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+-+2y+=+9 ALIGN=MIDDLE  ALT=\"4x+-+2y+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" > true @ <b><big>(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2F6 ALIGN=MIDDLE  ALT=\"11%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%285%2F6%29 ALIGN=MIDDLE  ALT=\"-%285%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)</b></big>?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2811%2F6%29-2%28-%285%2F6%29%29=%2844%2F6%29%2B%2810%2F6%29=54%2F6=9 ALIGN=MIDDLE  ALT=\"4%2811%2F6%29-2%28-%285%2F6%29%29=%2844%2F6%29%2B%2810%2F6%29=54%2F6=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Check.\r<BR>\n" );
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document.write( "Second, graphically:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C+%28x%2F2%29-%287%2F4%29%2C+2x-%289%2F2%29%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C+%28x%2F2%29-%287%2F4%29%2C+2x-%289%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note that the lines intersect at a point a little less than 2 [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2F6 ALIGN=MIDDLE  ALT=\"11%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >] and a little more than -1 [<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%285%2F6%29 ALIGN=MIDDLE  ALT=\"-%285%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >]. </SPAN>\n" );
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