document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1135198>Question 1135198</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following equations simultaneously.<BR>\n" );
document.write( "5x - 6y + 4z = 15<BR>\n" );
document.write( "7x + 4y - 3z = 19<BR>\n" );
document.write( "2x + y + 6z = 46<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 #752778 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=752778\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "There are always dozens of different paths you can take to solve a system of equations like this.  One path I would not take is the one the other tutor showed, which leads you through a whole bunch of ugly fractions.<br><BR>\n" );
document.write( "In general, in my opinion, the easiest way to solve a system like this is to eliminate one variable at a time.  Eliminate one variable to give a system of two equations in two unknowns; then eliminate one of the remaining variables and solve for the other.  Then work back through your equations to solve for the other two variables.<br><BR>\n" );
document.write( "Even with that general method, there are multiple paths you can take.  Here is one....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x+-+6y+%2B+4z+=+15 ALIGN=MIDDLE  ALT=\"5x+-+6y+%2B+4z+=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x+%2B+4y+-+3z+=+19 ALIGN=MIDDLE  ALT=\"7x+%2B+4y+-+3z+=+19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+y+%2B+6z+=+46 ALIGN=MIDDLE  ALT=\"2x+%2B+y+%2B+6z+=+46\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The coefficients make it look as if eliminating either y or z first will be easiest; I chose to eliminate z.<br><BR>\n" );
document.write( "If I multiply the first equation by 3 and the second by 4, the coefficients of z will be 12 and -12; when I add the two resulting equations, z will be eliminated.<br><BR>\n" );
document.write( "Similarly if I multiply the second equation by 2 and add it to the third equation.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x-18y%2B12z+=+45 ALIGN=MIDDLE  ALT=\"15x-18y%2B12z+=+45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28x%2B16y-12z+=+76 ALIGN=MIDDLE  ALT=\"28x%2B16y-12z+=+76\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=43x-2y+=+121 ALIGN=MIDDLE  ALT=\"43x-2y+=+121\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [1]<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14x%2B8y-6z+=+38 ALIGN=MIDDLE  ALT=\"14x%2B8y-6z+=+38\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By%2B6z+=+46 ALIGN=MIDDLE  ALT=\"2x%2By%2B6z+=+46\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%2B9y+=+84 ALIGN=MIDDLE  ALT=\"16x%2B9y+=+84\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [2]<br><BR>\n" );
document.write( "Now eliminate one of the variables between equations [1] and [2].  Eliminating y looks easier; multiply the first equation by 9 and the second by 2 and add.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=387x-18y+=+1089 ALIGN=MIDDLE  ALT=\"387x-18y+=+1089\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32x%2B18y+=+168 ALIGN=MIDDLE  ALT=\"32x%2B18y+=+168\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=419x+=+1257 ALIGN=MIDDLE  ALT=\"419x+=+1257\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\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;\" >  [3]<br><BR>\n" );
document.write( "Now work backwards to solve for the other variables.<br><BR>\n" );
document.write( "Substitute [3] in [2]:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2A3%2B9y+=+84 ALIGN=MIDDLE  ALT=\"16%2A3%2B9y+=+84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48%2B9y+=+84 ALIGN=MIDDLE  ALT=\"48%2B9y+=+84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9y+=+36 ALIGN=MIDDLE  ALT=\"9y+=+36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+4 ALIGN=MIDDLE  ALT=\"y+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [4]<br><BR>\n" );
document.write( "And now substitute [3] and [4] in one of the original equations:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%283%29%2B4%2B6z+=+46 ALIGN=MIDDLE  ALT=\"2%283%29%2B4%2B6z+=+46\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2B6z+=+46 ALIGN=MIDDLE  ALT=\"10%2B6z+=+46\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6z+=+36 ALIGN=MIDDLE  ALT=\"6z+=+36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=6 ALIGN=MIDDLE  ALT=\"z=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [5]<br><BR>\n" );
document.write( "The solution is x=3, y=4, z=6. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );