document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=239168>Question 239168</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if i were to need to find the point that three lines share is it possible for there to be no solution?<BR>\n" );
document.write( "the problem is <BR>\n" );
document.write( "6x+2y+4z=2<BR>\n" );
document.write( "3x+4y-8z=-3<BR>\n" );
document.write( "-3x-6y+2z=5\r<BR>\n" );
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document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #175656 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=175656\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if i were to need to find the point that three lines share is it possible for there to be no solution?<BR>\n" );
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document.write( "Yes.  Each equation is a plane.  They may meet in a point, or in<BR>\n" );
document.write( "a line or not at all:<BR>\n" );
document.write( "------------------------<BR>\n" );
document.write( "Mske the 3rd equation the 1st:<BR>\n" );
document.write( "-3x-6y+2z=5  <BR>\n" );
document.write( "6x+2y+4z=2<BR>\n" );
document.write( "3x+4y-8z=-3<BR>\n" );
document.write( "------------------<BR>\n" );
document.write( "Add twice the 1st equation to the 2nd<BR>\n" );
document.write( "Add the 1st equation to the 3rd:<BR>\n" );
document.write( "-3x-6y+2z=5  <BR>\n" );
document.write( "0x-10y+8z=12<BR>\n" );
document.write( "0x-2y-6z=2<BR>\n" );
document.write( "----------------------<BR>\n" );
document.write( "Make the 3rd equation the 2nd:<BR>\n" );
document.write( "-3x-6y+2z=5<BR>\n" );
document.write( "0x-2y-6z=2<BR>\n" );
document.write( "0x-10y+8z=12<BR>\n" );
document.write( "----------------------<BR>\n" );
document.write( "Divide thru the 2nd by -2<BR>\n" );
document.write( "Divide thru the 3rd by 2:<BR>\n" );
document.write( "-3x-6y+2z=5<BR>\n" );
document.write( "0x+y+3z=-1<BR>\n" );
document.write( "0x-5y+4z=6<BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "Add 5 times the 2nd Eq. to the 3rd to get:<BR>\n" );
document.write( "-3x-6y+2z=5<BR>\n" );
document.write( "0x+y+3z=-1<BR>\n" );
document.write( "0x+0y+19z=1<BR>\n" );
document.write( "----------------------<BR>\n" );
document.write( "z = 1/19<BR>\n" );
document.write( "-------------------<BR>\n" );
document.write( "Substitute into the 2nd equation and solve for \"y\":<BR>\n" );
document.write( "y + 3(1/19) = (-19/19)<BR>\n" );
document.write( "y = -22/19<BR>\n" );
document.write( "-------------------<BR>\n" );
document.write( "Substitute into the 1st equation to solve for \"x\":<BR>\n" );
document.write( "-3x-6y+2z=5<BR>\n" );
document.write( "-3x -6(-22/19) +2(1/19) = 5<BR>\n" );
document.write( "-3x + 132/19 + 2/19 = 95/19<BR>\n" );
document.write( "-3x = -39/19<BR>\n" );
document.write( "x = 13/19<BR>\n" );
document.write( "==========================-<BR>\n" );
document.write( "Conclusion: The three planes meet in the <BR>\n" );
document.write( "point (13/19, -22/19, 1/19)<BR>\n" );
document.write( "=====================================<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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