document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=82042>Question 82042</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following system by addition. If a unique solution does not exist, state whether the system is inconsistent or dependent:\r<BR>\n" );
document.write( "\n" );
document.write( "x+5y=10<BR>\n" );
document.write( "-2x-10y=-20 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #58861 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=58861\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\n" );
document.write( "\n" );
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" );
document.write( "<TR><TD  ALIGN=LEFT><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Lets start with the given system of linear equations<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B5%2Ay=10 ALIGN=MIDDLE  ALT=\"1%2Ax%2B5%2Ay=10\"   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-10%2Ay=-20 ALIGN=MIDDLE  ALT=\"-2%2Ax-10%2Ay=-20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
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" );
document.write( "  <BR>\n" );
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=5%2Ay=10-1%2Ax ALIGN=MIDDLE  ALT=\"5%2Ay=10-1%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" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%2810-1%2Ax%29%2F5 ALIGN=MIDDLE  ALT=\"y=%2810-1%2Ax%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 5. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Which breaks down and reduces to <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2-%281%2F5%29%2Ax ALIGN=MIDDLE  ALT=\"y=2-%281%2F5%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=2-%281%2F5%29%2Ax ALIGN=MIDDLE  ALT=\"2-%281%2F5%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=2-%281%2F5%29%2Ax ALIGN=MIDDLE  ALT=\"2-%281%2F5%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=-2%2Ax%2B-10%2Ahighlight%28%282-%281%2F5%29%2Ax%29%29=-20 ALIGN=MIDDLE  ALT=\"-2%2Ax%2B-10%2Ahighlight%28%282-%281%2F5%29%2Ax%29%29=-20\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace y with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-%281%2F5%29%2Ax ALIGN=MIDDLE  ALT=\"2-%281%2F5%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=-2%2Ax-10%2A%282%29-10%28-1%2F5%29x=-20 ALIGN=MIDDLE  ALT=\"-2%2Ax-10%2A%282%29-10%28-1%2F5%29x=-20\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute -10 to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2-%281%2F5%29%2Ax ALIGN=MIDDLE  ALT=\"2-%281%2F5%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=-2%2Ax-20%2B%2810%2F5%29%2Ax=-20 ALIGN=MIDDLE  ALT=\"-2%2Ax-20%2B%2810%2F5%29%2Ax=-20\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Ax-20%2B2%2Ax=-20 ALIGN=MIDDLE  ALT=\"-2%2Ax-20%2B2%2Ax=-20\"   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=-2%2Ax%2B2%2Ax=-20%2B20 ALIGN=MIDDLE  ALT=\"-2%2Ax%2B2%2Ax=-20%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" >Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   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=-2%2Ax%2B2%2Ax=0 ALIGN=MIDDLE  ALT=\"-2%2Ax%2B2%2Ax=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the terms on the right side<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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" );
document.write( "  <BR>\n" );
document.write( "  If we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B5%2Ay=10 ALIGN=MIDDLE  ALT=\"1%2Ax%2B5%2Ay=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Ax-10%2Ay=-20 ALIGN=MIDDLE  ALT=\"-2%2Ax-10%2Ay=-20\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get <BR>\n" );
document.write( "  <BR>\n" );
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+%2810-1%2Ax%29%2F5%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%2810-1%2Ax%29%2F5%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B5%2Ay=10 ALIGN=MIDDLE  ALT=\"1%2Ax%2B5%2Ay=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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+%28-20--2%2Ax%29%2F-10+%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%28-20--2%2Ax%29%2F-10+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Ax-10%2Ay=-20 ALIGN=MIDDLE  ALT=\"-2%2Ax-10%2Ay=-20\"   DISABLED_style= \"max-width:90%; height:auto;\" > (hint: you may have to solve for y to graph these)<BR>\n" );
document.write( "  <BR>\n" );
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" );
document.write( "</TABLE>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );