document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=111364>Question 111364</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the system of equations to find the point they intersect. <BR>\n" );
document.write( "3x-6y=9<BR>\n" );
document.write( "x-2y=3<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 #81188 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=81188\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\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=3%2Ax-6%2Ay=9 ALIGN=MIDDLE  ALT=\"3%2Ax-6%2Ay=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax-2%2Ay=3 ALIGN=MIDDLE  ALT=\"1%2Ax-2%2Ay=3\"   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=-6%2Ay=9-3%2Ax ALIGN=MIDDLE  ALT=\"-6%2Ay=9-3%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" >Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Ax ALIGN=MIDDLE  ALT=\"3%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=%289-3%2Ax%29%2F-6 ALIGN=MIDDLE  ALT=\"y=%289-3%2Ax%29%2F-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by -6. <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=-3%2F2%2B%281%2F2%29%2Ax ALIGN=MIDDLE  ALT=\"y=-3%2F2%2B%281%2F2%29%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=-3%2F2%2B%281%2F2%29%2Ax ALIGN=MIDDLE  ALT=\"-3%2F2%2B%281%2F2%29%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=-3%2F2%2B%281%2F2%29%2Ax ALIGN=MIDDLE  ALT=\"-3%2F2%2B%281%2F2%29%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=1%2Ax%2B-2%2Ahighlight%28%28-3%2F2%2B%281%2F2%29%2Ax%29%29=3 ALIGN=MIDDLE  ALT=\"1%2Ax%2B-2%2Ahighlight%28%28-3%2F2%2B%281%2F2%29%2Ax%29%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace y with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F2%2B%281%2F2%29%2Ax ALIGN=MIDDLE  ALT=\"-3%2F2%2B%281%2F2%29%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=1%2Ax-2%2A%28-3%2F2%29-2%281%2F2%29x=3 ALIGN=MIDDLE  ALT=\"1%2Ax-2%2A%28-3%2F2%29-2%281%2F2%29x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute -2 to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F2%2B%281%2F2%29%2Ax ALIGN=MIDDLE  ALT=\"-3%2F2%2B%281%2F2%29%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=1%2Ax%2B6%2F2-%282%2F2%29%2Ax=3 ALIGN=MIDDLE  ALT=\"1%2Ax%2B6%2F2-%282%2F2%29%2Ax=3\"   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=1%2Ax%2B3-1%2Ax=3 ALIGN=MIDDLE  ALT=\"1%2Ax%2B3-1%2Ax=3\"   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=1%2Ax-1%2Ax=3-3 ALIGN=MIDDLE  ALT=\"1%2Ax-1%2Ax=3-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   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=1%2Ax-1%2Ax=0 ALIGN=MIDDLE  ALT=\"1%2Ax-1%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=3%2Ax-6%2Ay=9 ALIGN=MIDDLE  ALT=\"3%2Ax-6%2Ay=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax-2%2Ay=3 ALIGN=MIDDLE  ALT=\"1%2Ax-2%2Ay=3\"   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+%289-3%2Ax%29%2F-6%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%289-3%2Ax%29%2F-6%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Ax-6%2Ay=9 ALIGN=MIDDLE  ALT=\"3%2Ax-6%2Ay=9\"   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+%283-1%2Ax%29%2F-2+%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%283-1%2Ax%29%2F-2+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax-2%2Ay=3 ALIGN=MIDDLE  ALT=\"1%2Ax-2%2Ay=3\"   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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