document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1139578>Question 1139578</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve by substitution<BR>\n" );
document.write( "2x+3y+2z=18<BR>\n" );
document.write( "-4x+y-z=5<BR>\n" );
document.write( "2z=-10 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #760078 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=760078\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "We have these three equations<BR>\n" );
document.write( "2x+3y+2z = 18<BR>\n" );
document.write( "-4x+y-z = 5<BR>\n" );
document.write( "2z = -10<BR>\n" );
document.write( "which we'll label <font color=blue>equation (1)</font>, <font color=blue>equation (2)</font>, and <font color=blue>equation (3)</font> respectively in that exact order.\r<BR>\n" );
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document.write( "Solve <font color=blue>equation (3)</font> for z<BR>\n" );
document.write( "2z=-10 <BR>\n" );
document.write( "2z/2=-10/2  .... divide both sides by 2<BR>\n" );
document.write( "z = -5\r<BR>\n" );
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document.write( "Plug this into <font color=blue>equation (2)</font> and isolate y<BR>\n" );
document.write( "-4x+y-z = 5<BR>\n" );
document.write( "-4x+y-(-5) = 5 ... replace z with -5<BR>\n" );
document.write( "-4x+y+5 = 5<BR>\n" );
document.write( "-4x+y+5-5 = 5-5 .... subtract 5 from both sides<BR>\n" );
document.write( "-4x+y = 0<BR>\n" );
document.write( "-4x+y+4x = 0+4x .... add 4x to both sides<BR>\n" );
document.write( "y = 4x<BR>\n" );
document.write( "Call this <font color=blue>equation (4)</font> as we'll use it later.\r<BR>\n" );
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document.write( "Move to <font color=blue>equation (1)</font>. <BR>\n" );
document.write( "Plug in z = -5 and also y = 4x which was <font color=blue>equation (4)</font> we found earlier<BR>\n" );
document.write( "2x+3y+2z = 18<BR>\n" );
document.write( "2x+3y+2(-5) = 18 .... replace every z with -5<BR>\n" );
document.write( "2x+3y-10 = 18<BR>\n" );
document.write( "2x+3(4x)-10 = 18 ... replace every y with (4x), since y = 4x from  <font color=blue>equation (4)</font><BR>\n" );
document.write( "2x+12x-10 = 18<BR>\n" );
document.write( "2x+12x-10+10 = 18+10 ... add 10 to both sides<BR>\n" );
document.write( "14x = 28<BR>\n" );
document.write( "14x/14 = 28/14 ... divide both sides by 14<BR>\n" );
document.write( "x = 2\r<BR>\n" );
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document.write( "So far we know that<BR>\n" );
document.write( "x = 2<BR>\n" );
document.write( "z = -5\r<BR>\n" );
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document.write( "We can plug those values into any equation that has y in it, and then solve for y. So that constitutes everything but <font color=blue>equation (3)</font>. I'll use <font color=blue>equation (4)</font> since we isolated y here. It's easiest to use an equation with the variable already isolated.\r<BR>\n" );
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document.write( "y = 4*x <BR>\n" );
document.write( "y = 4*2 ... replace every x with 2<BR>\n" );
document.write( "y = 8\r<BR>\n" );
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document.write( "------------------------------------------------------------------------<BR>\n" );
document.write( "Final Answer:\r<BR>\n" );
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document.write( "The solution as an ordered triple is <font color=red size=4>(x,y,z) = (2, 8, -5)</font><BR>\n" );
document.write( "Which breaks down to <font color=red size=4>x = 2, y = 8 and z = -5</font>\r<BR>\n" );
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document.write( "Visually, each of the three original equations plots out a plane if you're in a 3D coordinate system. Those three planes will intersect at the location (2, 8, -5). Think of 3 pieces of flat paper all meeting up at the same spot. \r<BR>\n" );
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document.write( "To verify the answer, you need to replace the variables with their proper corresponding values. Afterward, simplify the equation. You should get the same value on both sides. Getting the same value on both sides says to the reader \"this equation is true\". To fully verify a system of equations, all of the equations must be true for the same variable values. I'll leave the verification step up for you to try out. <BR>\n" );
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document.write( " </SPAN>\n" );
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