document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=88887>Question 88887</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For equations that have two variables, the solutions come in \"ordered pairs\". In most cases there are an infinite number of solutions. Explain how you could use the graph of an equation (function) to obtain some of these solutions. \r<BR>\n" );
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document.write( "please explain this to me I am so terribly confused by the question </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #64584 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=64584\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Say you had the system:\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" );
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document.write( "Now lets solve by substitution:\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solving_linear_system_by_substitution.solver>Solving a linear system of equations by subsitution</A></TD></TR>\n" );
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document.write( "  Lets start with the given system of linear equations<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B1%2Ay=2 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%2Ay=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B2%2Ay=4 ALIGN=MIDDLE  ALT=\"2%2Ax%2B2%2Ay=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.<BR>\n" );
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document.write( "  Solve for y for the first equation <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ay=2-1%2Ax ALIGN=MIDDLE  ALT=\"1%2Ay=2-1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" >Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax ALIGN=MIDDLE  ALT=\"1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282-1%2Ax%29 ALIGN=MIDDLE  ALT=\"y=%282-1%2Ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 1. <BR>\n" );
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document.write( "  Which breaks down and reduces to <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2-1%2Ax ALIGN=MIDDLE  ALT=\"y=2-1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now we've fully isolated y<BR>\n" );
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document.write( "  Since y equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-1%2Ax ALIGN=MIDDLE  ALT=\"2-1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can substitute the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-1%2Ax ALIGN=MIDDLE  ALT=\"2-1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > into y of the 2nd equation. This will eliminate y so we can solve for x.<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B2%2Ahighlight%28%282-1%2Ax%29%29=4 ALIGN=MIDDLE  ALT=\"2%2Ax%2B2%2Ahighlight%28%282-1%2Ax%29%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace y with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-1%2Ax ALIGN=MIDDLE  ALT=\"2-1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since this eliminates y, we can now solve for x.<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B2%2A%282%29%2B2%28-1%29x=4 ALIGN=MIDDLE  ALT=\"2%2Ax%2B2%2A%282%29%2B2%28-1%29x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute 2 to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-1%2Ax ALIGN=MIDDLE  ALT=\"2-1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B4-2%2Ax=4 ALIGN=MIDDLE  ALT=\"2%2Ax%2B4-2%2Ax=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B4-2%2Ax=4 ALIGN=MIDDLE  ALT=\"2%2Ax%2B4-2%2Ax=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce any fractions<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax-2%2Ax=4-4 ALIGN=MIDDLE  ALT=\"2%2Ax-2%2Ax=4-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax-2%2Ax=0 ALIGN=MIDDLE  ALT=\"2%2Ax-2%2Ax=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the terms on the right side<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2Ax=0 ALIGN=MIDDLE  ALT=\"0%2Ax=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now combine the terms on the left side.  <BR>\n" );
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;\" > Since this expression is true for any x, we have an identity. <BR>\n" );
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document.write( "  So there are an infinite number solutions. The simple reason is the 2 equations represent 2 lines that overlap each other. So they intersect each other at an infinite number of points.<BR>\n" );
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document.write( "  If we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B1%2Ay=2 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%2Ay=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B2%2Ay=4 ALIGN=MIDDLE  ALT=\"2%2Ax%2B2%2Ay=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%282-1%2Ax%29%2F1%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%282-1%2Ax%29%2F1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B1%2Ay=2 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%2Ay=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%284-2%2Ax%29%2F2+%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%284-2%2Ax%29%2F2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B2%2Ay=4 ALIGN=MIDDLE  ALT=\"2%2Ax%2B2%2Ay=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (hint: you may have to solve for y to graph these)<BR>\n" );
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document.write( "  we can see that these two lines are the same. So this system is <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/coordinate/lessons/Types-of-systems-inconsistent-dependent-independent.lesson>dependent</a></TD></TR>\n" );
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document.write( "So basically any x value will work as a solution. So this means there are an infinite number of solutions.\r<BR>\n" );
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