document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1156489>Question 1156489</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation:\r<BR>\n" );
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document.write( "Solve the system of equations.\r<BR>\n" );
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document.write( "3x−4y−7z = −6<BR>\n" );
document.write( "2x+3y−5z = 1\r<BR>\n" );
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document.write( "(If the system is dependent, enter a general solution in terms of c.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #779201 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13330)</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=779201\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "There are three variables and only two equations.  Therefore we can't get a single solution; we can only get a family of solutions in terms of some parameter.<br><BR>\n" );
document.write( "The process is straightforward, although the calculations are generally a bit ugly.<br><BR>\n" );
document.write( "(1) Use elimination to reduce the system of 2 equations and 3 unknowns to a system of 1 equation with 2 unknowns.<BR>\n" );
document.write( "(2) Solve that single equation for one variable in terms of the other.<BR>\n" );
document.write( "(3) Substitute into either of the original equations to find an expression for the third variable.<br><BR>\n" );
document.write( "(1) I chose to eliminate x: multiply the first equation by 2 and the second equation by -3 and add:<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "  6x - 8y - 14z = -12\r\n" );
document.write( " -6x - 9y + 15z = -3\r\n" );
document.write( " ---------------------\r\n" );
document.write( "     -17y +  z  = -15</PRE><BR>\n" );
document.write( "Solve for z in terms of y:<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "             z = 17y - 15</PRE><BR>\n" );
document.write( "(3) Substitute y for y and 17y-15 for z in one of the original equations to find x in terms of y:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B3y-5%2817y-15%29+=+1 ALIGN=MIDDLE  ALT=\"2x%2B3y-5%2817y-15%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B3y-85y%2B75+=+1 ALIGN=MIDDLE  ALT=\"2x%2B3y-85y%2B75+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+=+82y-74 ALIGN=MIDDLE  ALT=\"2x+=+82y-74\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+41y-37 ALIGN=MIDDLE  ALT=\"x+=+41y-37\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER:<BR>\n" );
document.write( "x = 41y-37; y = y; z = 17y - 15<br><BR>\n" );
document.write( "Choose any number for the \"parameter\" y and use it to find values for x and z.  Those values of x, y, and z will satisfy both of the given equations.<br><BR>\n" );
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