document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=129818>Question 129818</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0078213479/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> can you assist me to solved this linear equation by graphing<BR>\n" );
document.write( " x+2y=5<BR>\n" );
document.write( " x-y=2\r<BR>\n" );
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document.write( "how do I determin wheather the following has one solution, no solution or many solution. : y=-x+2<BR>\n" );
document.write( "            3x+3y=6\r<BR>\n" );
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document.write( "use  substitution to solved:  2x+y=7<BR>\n" );
document.write( "                              5x-2y=7\r<BR>\n" );
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document.write( "Thank you for your assistant </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #94799 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=94799\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1) To solve a system by graphing, you need to graph each of the lines on the same coordinate system and then find the point of intersection.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-x%2F2%2B5%2F2%2Cx-2%29%2C%0D%0Acircle%283%2C1%2C.1%29%2C%0D%0Alocate%283.2%2C.8%2CS%283%2C1%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "2) First, put both of the equations in standard form:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%2BBy=C ALIGN=MIDDLE  ALT=\"Ax%2BBy=C\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-x%2B2 ALIGN=MIDDLE  ALT=\"y=-x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:  Add x to both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=2 ALIGN=MIDDLE  ALT=\"x%2By=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The second equation is already in standard form, so your two equations are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=2 ALIGN=MIDDLE  ALT=\"x%2By=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B3y=6 ALIGN=MIDDLE  ALT=\"3x%2B3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Compare the coefficients on one of the variables.  The coefficients on the x variable are 1 and 3.  If you multiply the first equation by -3, then the coefficients on x will be additive inverses, namely -3 and 3.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x-3y=-6 ALIGN=MIDDLE  ALT=\"-3x-3y=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B3y=6 ALIGN=MIDDLE  ALT=\"3x%2B3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now add the two equations term by term:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0x+%2B+0y+=+0 ALIGN=MIDDLE  ALT=\"0x+%2B+0y+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We got the result that <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;\" >, a statement that is always true.  That means that ANY ordered pair that will satisfy one of the equations will also satisfy the other, hence the solution set for this system has an infinite number of elements.\r<BR>\n" );
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document.write( "In contrast, what if we had a similar system, say:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B3y=6 ALIGN=MIDDLE  ALT=\"3x%2B3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's do the same process, multiply equation 1 by -3:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B3y=6 ALIGN=MIDDLE  ALT=\"3x%2B3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And add the two equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0x+%2B+0y+=+-3 ALIGN=MIDDLE  ALT=\"0x+%2B+0y+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we have the always false statement <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-3 ALIGN=MIDDLE  ALT=\"0+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  In this case there is no ordered pair (x,y) that satisfies both equation simultaneously.  The solution set to this system is therefore empty.\r<BR>\n" );
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document.write( "Graphically speaking, the problem as stated gave you two different forms of the equation of the same line, so all of the points of one are in common with all of the points of the other.  In my modified example, we have two equations that represent parallel lines.  Parallel lines have no point of intersection, so there is no solution to the system.\r<BR>\n" );
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document.write( "Original equations:\r<BR>\n" );
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document.write( "You can only see one line on the graph because the two lines are the same line:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-x%2B2%2C-x%2B2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Modified equations:\r<BR>\n" );
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document.write( "You can see that the two lines never intersect:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-x%2B2%2C-x%2B3%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "3)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By=7 ALIGN=MIDDLE  ALT=\"2x%2By=7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-2y=7 ALIGN=MIDDLE  ALT=\"5x-2y=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To solve by substitution, the first step is to solve (re-arrange) one of the equations to get an expression for one of the variables in terms of the other.  Let's perform this process on the first equation and solve for y:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By=7 ALIGN=MIDDLE  ALT=\"2x%2By=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >:  Add -2x to both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%2B7 ALIGN=MIDDLE  ALT=\"y=-2x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we can take this expression for y and substitute it (put it in place of) the y 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=5x-2y=7 ALIGN=MIDDLE  ALT=\"5x-2y=7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-2%28-2x%2B7%29=7 ALIGN=MIDDLE  ALT=\"5x-2%28-2x%2B7%29=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >:  Distribute the -2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B4x-14=7 ALIGN=MIDDLE  ALT=\"5x%2B4x-14=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >:     Add 14 to both sides and combine like terms<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x=21 ALIGN=MIDDLE  ALT=\"9x=21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=21%2F9=7%2F3 ALIGN=MIDDLE  ALT=\"x=21%2F9=7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now that we have a value for x, we can substitute that into either of the original equations:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%287%2F3%29%2By=7 ALIGN=MIDDLE  ALT=\"2%287%2F3%29%2By=7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14%2F3%2By=7 ALIGN=MIDDLE  ALT=\"14%2F3%2By=7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=7-14%2F3 ALIGN=MIDDLE  ALT=\"y=7-14%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%2821-14%29%2F3 ALIGN=MIDDLE  ALT=\"y=%2821-14%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=7%2F3 ALIGN=MIDDLE  ALT=\"y=7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And the solution set is the ordered pair:  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F3 ALIGN=MIDDLE  ALT=\"7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F3 ALIGN=MIDDLE  ALT=\"7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Check<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-5%2C5%2C-5%2C5%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-2x%2B7%2C5x%2F2-7%2F2%29%2C%0D%0Acircle%287%2F3%2C7%2F3%2C.1%29%2C%0D%0Alocate%28%287%2F3%29%2B.2%2C%287%2F3%29-.2%2CS%287%2F3%2C7%2F3%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Super-double-plus extra credit:<BR>\n" );
document.write( "On the graph above, which line is a graph of which equation?  Triple credit if you do it in your head just looking at the equations. </SPAN>\n" );
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