document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=207149>Question 207149</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the system by the addition method.  Determine whether the equations are independent, dependent, or inconsistent.<BR>\n" );
document.write( "3x - 2y = -7<BR>\n" );
document.write( "x – 5y = 2<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #156596 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=156596\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
document.write( "\n" );
document.write( "Start with the given system of equations:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x-2y=-7%2Cx-5y=2%29 ALIGN=MIDDLE  ALT=\"system%283x-2y=-7%2Cx-5y=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%28x-5y%29=-3%282%29 ALIGN=MIDDLE  ALT=\"-3%28x-5y%29=-3%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the both sides of the second equation by -3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%2B15y=-6 ALIGN=MIDDLE  ALT=\"-3x%2B15y=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute and multiply.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we have the new system of equations:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x-2y=-7%2C-3x%2B15y=-6%29 ALIGN=MIDDLE  ALT=\"system%283x-2y=-7%2C-3x%2B15y=-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x-2y%29%2B%28-3x%2B15y%29=%28-7%29%2B%28-6%29 ALIGN=MIDDLE  ALT=\"%283x-2y%29%2B%28-3x%2B15y%29=%28-7%29%2B%28-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B-3x%29%2B%28-2y%2B15y%29=-7%2B-6 ALIGN=MIDDLE  ALT=\"%283x%2B-3x%29%2B%28-2y%2B15y%29=-7%2B-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0x%2B13y=-13 ALIGN=MIDDLE  ALT=\"0x%2B13y=-13\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13y=-13 ALIGN=MIDDLE  ALT=\"13y=-13\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-13%29%2F%2813%29 ALIGN=MIDDLE  ALT=\"y=%28-13%29%2F%2813%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1 ALIGN=MIDDLE  ALT=\"y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-2y=-7 ALIGN=MIDDLE  ALT=\"3x-2y=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now go back to the first equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-2%28-1%29=-7 ALIGN=MIDDLE  ALT=\"3x-2%28-1%29=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1 ALIGN=MIDDLE  ALT=\"y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B2=-7 ALIGN=MIDDLE  ALT=\"3x%2B2=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=-7-2 ALIGN=MIDDLE  ALT=\"3x=-7-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=-9 ALIGN=MIDDLE  ALT=\"3x=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-9%29%2F%283%29 ALIGN=MIDDLE  ALT=\"x=%28-9%29%2F%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1 ALIGN=MIDDLE  ALT=\"y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Which form the ordered pair <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(-3,-1\right)\">.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means that the system is consistent and independent.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice when we graph the equations, we see that they intersect at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(-3,-1\right)\">. So this visually verifies our answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-13%2C7%2C-11%2C9%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28500%2C500%2C-13%2C7%2C-11%2C9%2C%28-7-3x%29%2F%28-2%29%2C%282-x%29%2F%28-5%29%29%2C%0D%0Acircle%28-3%2C-1%2C0.05%29%2C%0D%0Acircle%28-3%2C-1%2C0.08%29%2C%0D%0Acircle%28-3%2C-1%2C0.10%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-2y=-7 ALIGN=MIDDLE  ALT=\"3x-2y=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-5y=2 ALIGN=MIDDLE  ALT=\"x-5y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green) \r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );