document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=335524>Question 335524</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3x+y+z=5<BR>\n" );
document.write( "4x+5y-z=-8<BR>\n" );
document.write( "10x+7y+z=2<BR>\n" );
document.write( "Solve by Gauss-Jordan method </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #240658 by </B><A HREF=/tutors/aboutme.mpl?userid=CharlesG2><B>CharlesG2(834)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CharlesG2><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=240658\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 3x+y+z=5<BR>\n" );
document.write( "4x+5y-z=-8<BR>\n" );
document.write( "10x+7y+z=2<BR>\n" );
document.write( "Solve by Gauss-Jordan method\r<BR>\n" );
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document.write( "what we are going to do: use arithmetic operations to eliminate variables and put x, y, and z in a diagonal opposite their solutions\r<BR>\n" );
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document.write( " 3x +  y + z =  5<BR>\n" );
document.write( " 4x + 5y - z = -8 (add to 1st equation)<BR>\n" );
document.write( "10x + 7y + z =  2<BR>\n" );
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document.write( " 7x + 6y + 0z = -3<BR>\n" );
document.write( " 4x + 5y -  z = -8<BR>\n" );
document.write( "10x + 7y +  z =  2 (add to 2nd equation)\r<BR>\n" );
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document.write( " 7x +  6y + 0z = -3 (multiply both sides by -2)<BR>\n" );
document.write( "14x + 12y + 0z = -6<BR>\n" );
document.write( "10x +  7y +  z =  2\r<BR>\n" );
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document.write( "-14x - 12y + 0z =  6<BR>\n" );
document.write( " 14x + 12y + 0z = -6 (add to 1st equation)<BR>\n" );
document.write( " 10x +  7y +  z =  2\r<BR>\n" );
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document.write( "  0x +  0y + 0z =  0<BR>\n" );
document.write( " 14x + 12y + 0z = -6 (2 variables here)<BR>\n" );
document.write( " 10x +  7y +  z =  2 (3 variables here, we have a problem)\r<BR>\n" );
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document.write( "we still have 3 variables to solve for\r<BR>\n" );
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document.write( "we just lost one equation\r<BR>\n" );
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document.write( "we need 3 equations to solve for 3 variables\r<BR>\n" );
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document.write( "this has an infinite number of solutions\r<BR>\n" );
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document.write( "cases of having fewer equations than unknowns usually have an infinite number of solutions\r<BR>\n" );
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document.write( "these are referred to as underdetermined systems\r<BR>\n" );
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document.write( "this problem has an infinite number of solutions\r<BR>\n" );
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document.write( "if anyone finds an error in what I have done here please let me know\r<BR>\n" );
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document.write( "-----------------------\r<BR>\n" );
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document.write( "REVISITING PROBLEM\r<BR>\n" );
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document.write( "  0x +  0y + 0z =  0<BR>\n" );
document.write( " 14x + 12y + 0z = -6 (multiply both sides by 5)<BR>\n" );
document.write( " 10x +  7y +  z =  2 (multiply both sides by -7)\r<BR>\n" );
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document.write( "  0x +   0y + 0z =   0<BR>\n" );
document.write( " 70x +  60y + 0z = -30 (add to 3rd equation)<BR>\n" );
document.write( "-70x -  49y - 7z = -14\r<BR>\n" );
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document.write( "  0x +   0y + 0z =   0<BR>\n" );
document.write( " 70x +  60y + 0z = -30<BR>\n" );
document.write( "  0x +  11y - 7z = -44\r<BR>\n" );
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document.write( "from 2nd equation:<BR>\n" );
document.write( "60y = -70x - 30<BR>\n" );
document.write( " 6y = - 7x -  3<BR>\n" );
document.write( "  y = (-7/6)x - 3/6<BR>\n" );
document.write( "  y = (-7/6)x - 1/2\r<BR>\n" );
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document.write( "check on 2nd equation:<BR>\n" );
document.write( "70x + 60(-7/6)x - 60(1/2) = -30<BR>\n" );
document.write( "70x + 10(-7)x - 30 = -30<BR>\n" );
document.write( "70x - 70x - 30 = -30<BR>\n" );
document.write( "0x - 30 = -30<BR>\n" );
document.write( "-30 = -30\r<BR>\n" );
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document.write( "plug into 3rd equation:\r<BR>\n" );
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document.write( "0x + 11(-7/6)x - 11(1/2) - 7z = -44<BR>\n" );
document.write( "0x - (77/6)x - (11/2) - 7z = -44<BR>\n" );
document.write( "0x - (154/6)x - 11 - 14z = -88 (multiplied both sides by 2)<BR>\n" );
document.write( "- (154/6)x - 14z = -77<BR>\n" );
document.write( "-154x - 84z = -462 (multiplied both sides by 6)\r<BR>\n" );
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document.write( "-84z = 154x - 462<BR>\n" );
document.write( " 84z = -154x + 462 (divided both sides by -1)<BR>\n" );
document.write( "   z = (-154/84)x + (462/84)<BR>\n" );
document.write( "   z = (-77/42)x + (231/42) (simplified both fractions by 2/2)<BR>\n" );
document.write( "   z = (-11/6)x + (33/6) (simplified both fractions by 7/7)<BR>\n" );
document.write( "   z = (-11/6)x + (11/2) (simplified 2nd fraction by 3/3)\r<BR>\n" );
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document.write( "now we got x = ? <BR>\n" );
document.write( "y = (-7/6)x - 1/2<BR>\n" );
document.write( "(a line in x-y plane with slope -7/6 and y-intercept -1/2)<BR>\n" );
document.write( "z = (-11/6)x + (11/2) <BR>\n" );
document.write( "(a line in x-z plane with slope -11/6 and z-intercept 11/2)\r<BR>\n" );
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document.write( "lets plug these into original equations:\r<BR>\n" );
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document.write( "  3x +  y + z =  5<BR>\n" );
document.write( "  4x + 5y - z = -8<BR>\n" );
document.write( " 10x + 7y + z =  2\r<BR>\n" );
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document.write( "  3x +   (-7/6)x - 1/2  + (-11/6)x + (11/2)     =  5<BR>\n" );
document.write( "  4x + 5((-7/6)x - 1/2) -((-11/6)x + (11/2)) = -8<BR>\n" );
document.write( " 10x + 7((-7/6)x - 1/2) + (-11/6)x + (11/2)  =  2\r<BR>\n" );
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document.write( "  3x - (7/6)x - (1/2) - (11/6)x + (11/2)  =  5<BR>\n" );
document.write( "  4x - (35/6)x - (5/2) + (11/6)x - (11/2) = -8<BR>\n" );
document.write( " 10x - (49/6)x - (7/2) - (11/6)x + (11/2) =  2\r<BR>\n" );
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document.write( "  3x - (18/6)x + (10/2) =  5<BR>\n" );
document.write( "  4x - (24/6)x - (16/2) = -8<BR>\n" );
document.write( " 10x - (60/6)x + (4/2)  =  2\r<BR>\n" );
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document.write( "  3x -  3x + 5 =  5<BR>\n" );
document.write( "  4x -  4x - 8 = -8<BR>\n" );
document.write( " 10x - 10x + 2 =  2\r<BR>\n" );
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document.write( "no solution for x<BR>\n" );
document.write( "y = (-7/6)x - 1/2<BR>\n" );
document.write( "(a line in x-y plane with slope -7/6 and y-intercept -1/2)<BR>\n" );
document.write( "z = (-11/6)x + (11/2) <BR>\n" );
document.write( "(a line in x-z plane with slope -11/6 and z-intercept 11/2)\r<BR>\n" );
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document.write( "again, if you find an error in what I have done please let me know\r<BR>\n" );
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document.write( " </SPAN>\n" );
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