document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=86048>Question 86048</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve each of the following systems by substitution.\r<BR>\n" );
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document.write( "8x-4y=16<BR>\n" );
document.write( "y=2x-4\r<BR>\n" );
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document.write( "4x-12y=5<BR>\n" );
document.write( "-x+ 3y=-1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62182 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=62182\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Lets start with the given system of linear equations\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2Ax-4%2Ay=16 ALIGN=MIDDLE  ALT=\"8%2Ax-4%2Ay=16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x-4 ALIGN=MIDDLE  ALT=\"y=2x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since y equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-4 ALIGN=MIDDLE  ALT=\"2x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can substitute the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-4 ALIGN=MIDDLE  ALT=\"2x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > into y of the 1st equation. This will eliminate y so we can solve for x.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2Ax%2B-4%2Ahighlight%28%282x-4%29%29=16 ALIGN=MIDDLE  ALT=\"8%2Ax%2B-4%2Ahighlight%28%282x-4%29%29=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace y with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-4 ALIGN=MIDDLE  ALT=\"2x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since this eliminates y, we can now solve for x.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2Ax-4%2A%282x+%29-4%28-4+%29x=16 ALIGN=MIDDLE  ALT=\"8%2Ax-4%2A%282x+%29-4%28-4+%29x=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute -4 to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-4 ALIGN=MIDDLE  ALT=\"2x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2Ax-8x%2B+16=16 ALIGN=MIDDLE  ALT=\"8%2Ax-8x%2B+16=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2Ax-8%2Ax=16-16+ ALIGN=MIDDLE  ALT=\"8%2Ax-8%2Ax=16-16+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2Ax-8%2Ax=0+ ALIGN=MIDDLE  ALT=\"8%2Ax-8%2Ax=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the terms on the right side\r<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.\r<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.\r<BR>\n" );
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document.write( "If we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%2By=-4 ALIGN=MIDDLE  ALT=\"-2x%2By=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x-4 ALIGN=MIDDLE  ALT=\"y=2x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get\r<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+%28-4--2%2Ax%29%2F1%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%28-4--2%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=-2%2Ax%2B1%2Ay=-4 ALIGN=MIDDLE  ALT=\"-2%2Ax%2B1%2Ay=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<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>\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( "  <BR>\n" );
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=4%2Ax-12%2Ay=5 ALIGN=MIDDLE  ALT=\"4%2Ax-12%2Ay=5\"   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%2B3%2Ay=-1 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ay=-1\"   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" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12%2Ay=5-4%2Ax ALIGN=MIDDLE  ALT=\"-12%2Ay=5-4%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" >Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ax ALIGN=MIDDLE  ALT=\"4%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%285-4%2Ax%29%2F-12 ALIGN=MIDDLE  ALT=\"y=%285-4%2Ax%29%2F-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by -12. <BR>\n" );
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document.write( "  Which breaks down and reduces to <BR>\n" );
document.write( "  <BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5%2F12%2B%281%2F3%29%2Ax ALIGN=MIDDLE  ALT=\"y=-5%2F12%2B%281%2F3%29%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now we've fully isolated y<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Since y equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F12%2B%281%2F3%29%2Ax ALIGN=MIDDLE  ALT=\"-5%2F12%2B%281%2F3%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=-5%2F12%2B%281%2F3%29%2Ax ALIGN=MIDDLE  ALT=\"-5%2F12%2B%281%2F3%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" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B3%2Ahighlight%28%28-5%2F12%2B%281%2F3%29%2Ax%29%29=-1 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ahighlight%28%28-5%2F12%2B%281%2F3%29%2Ax%29%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace y with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F12%2B%281%2F3%29%2Ax ALIGN=MIDDLE  ALT=\"-5%2F12%2B%281%2F3%29%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since this eliminates y, we can now solve for x.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B3%2A%28-5%2F12%29%2B3%281%2F3%29x=-1 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2A%28-5%2F12%29%2B3%281%2F3%29x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute 3 to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F12%2B%281%2F3%29%2Ax ALIGN=MIDDLE  ALT=\"-5%2F12%2B%281%2F3%29%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax-15%2F12%2B%283%2F3%29%2Ax=-1 ALIGN=MIDDLE  ALT=\"-1%2Ax-15%2F12%2B%283%2F3%29%2Ax=-1\"   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-5%2F4%2B1%2Ax=-1 ALIGN=MIDDLE  ALT=\"-1%2Ax-5%2F4%2B1%2Ax=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce any fractions<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B1%2Ax=-1%2B5%2F4 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B1%2Ax=-1%2B5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F4 ALIGN=MIDDLE  ALT=\"5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B1%2Ax=-4%2F4%2B5%2F4 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B1%2Ax=-4%2F4%2B5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Make -1 into a fraction with a denominator of 4<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B1%2Ax=1%2F4 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B1%2Ax=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the terms on the right side<BR>\n" );
document.write( "  <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2Ax=1%2F4 ALIGN=MIDDLE  ALT=\"0%2Ax=1%2F4\"   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%2F1=1%2F4 ALIGN=MIDDLE  ALT=\"0%2F1=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since this expression is not true, we have an inconsistency. <BR>\n" );
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document.write( "  So there are no solutions. The simple reason is the 2 equations represent 2 parallel lines that will never intersect. Since no intersections occur, no solutions exist.<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+%285-4%2Ax%29%2F-12%2C+%28-1--1%2Ax%29%2F3+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ax-12%2Ay=5 ALIGN=MIDDLE  ALT=\"4%2Ax-12%2Ay=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B3%2Ay=-1 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ay=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green) (hint: you may have to solve for y to graph these)<BR>\n" );
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document.write( "  and we can see that the two equations are parallel and will never intersect. So this system is <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/coordinate/lessons/Types-of-systems-inconsistent-dependent-independent.lesson>inconsistent</a></TD></TR>\n" );
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