document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184020>Question 1184020</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the following systems of equations<BR>\n" );
document.write( "2x + y+ z = 12<BR>\n" );
document.write( "6x + 5 y - 3z = 6<BR>\n" );
document.write( "4x - y + 3z = 5<BR>\n" );
document.write( "Solve for the x,y and z using:<BR>\n" );
document.write( "(i) Gaussian elimination<BR>\n" );
document.write( "(ii) Gauss- Jordan elimination<BR>\n" );
document.write( "(iii) Use Sarrus` rule to find determinant of the equation above<BR>\n" );
document.write( "(iv) Cramer`s rule </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814559 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814559\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Given the following systems of equations\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+y%2B+z+=+12 ALIGN=MIDDLE  ALT=\"2x+%2B+y%2B+z+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x+%2B+5+y+-+3z+=+6 ALIGN=MIDDLE  ALT=\"6x+%2B+5+y+-+3z+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+-+y+%2B+3z+=+5 ALIGN=MIDDLE  ALT=\"4x+-+y+%2B+3z+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solve for the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+ ALIGN=MIDDLE  ALT=\"y+\"   DISABLED_style= \"max-width:90%; height:auto;\" >and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" > using:\r<BR>\n" );
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document.write( "(i) Gaussian elimination\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C2%2C%091%2C%091%2C%0912%2C%0D%0A6%2C%095%2C%09-3%2C%096%2C%0D%0A4%2C%09-1%2C%093%2C%095%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "R1 / 2 → R1 (divide the 1 row by 2)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%090.5%2C%090.5%2C%096%2C%0D%0A6%2C%095%2C%09-3%2C%096%2C%0D%0A4%2C%09-1%2C%093%2C%095%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "R2 - 6 R1 → R2 (multiply 1 row by 6 and subtract it from 2 row); R3 - 4 R1 → R3 (multiply 1 row by 4 and subtract it from 3 row)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%090.5%2C%090.5%2C%096%2C%0D%0A0%2C%092%2C%09-6%2C%09-30%2C%0D%0A0%2C%09-3%2C%091%2C%09-19%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "R2 / 2 → R2 (divide the 2 row by 2)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%090.5%2C%090.5%2C%096%2C%0D%0A0%2C%091%2C%09-3%2C%09-15%2C%0D%0A0%2C%09-3%2C%091%2C%09-19%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "R1 - 0.5 R2 → R1 (multiply 2 row by 0.5 and subtract it from 1 row); R3 + 3 R2 → R3 (multiply 2 row by 3 and add it to 3 row)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%090%2C%092%2C%0913.5%2C%0D%0A0%2C%091%2C%09-3%2C%09-15%2C%0D%0A0%2C%090%2C%09-8%2C%09-64%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "R3 / -8 → R3 (divide the 3 row by -8)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%090%2C%092%2C%0913.5%2C%0D%0A0%2C%091%2C%09-3%2C%09-15%2C%0D%0A0%2C%090%2C%091%2C%098%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "R1 - 2 R3 → R1 (multiply 3 row by 2 and subtract it from 1 row); R2 + 3 R3 → R2 (multiply 3 row by 3 and add it to 2 row)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%090%2C%090%2C%09-2.5%2C%0D%0A0%2C%091%2C%090%2C%099%2C%0D%0A0%2C%090%2C%091%2C%098%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2.5 ALIGN=MIDDLE  ALT=\"x+=+-2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+9 ALIGN=MIDDLE  ALT=\"y+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+8 ALIGN=MIDDLE  ALT=\"z+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(ii) Gauss- Jordan elimination\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C%0D%0A2%2C%091%2C%091%2C%0912%2C%0D%0A6%2C%095%2C%09-3%2C%096%2C%0D%0A4%2C%09-1%2C%093%2C%095%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Make the pivot in the 1st column by dividing the 1st row by 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%091%2F2%2C%091%2F2%2C%096%2C%0D%0A6%2C%095%2C%09-3%2C%096%2C%0D%0A4%2C%09-1%2C%093%2C%095%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Eliminate the 1st column\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%091%2F2%2C%091%2F2%2C%096%2C%0D%0A0%2C%092%2C%09-6%2C%09-30%2C%0D%0A0%2C%09-3%2C%091%2C%09-19%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Make the pivot in the 2nd column by dividing the 2nd row by 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%091%2F2%2C%091%2F2%2C%096%2C%0D%0A0%2C%091%2C%09-3%2C%09-15%2C%0D%0A0%2C%09-3%2C%091%2C%09-19%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Eliminate the 2nd column\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%090%2C%092%2C%0927%2F2%2C%0D%0A0%2C%091%2C%09-3%2C%09-15%2C%0D%0A0%2C%090%2C%09-8%2C%09-64%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Make the pivot in the 3rd column by dividing the 3rd row by -8\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C1%2C%090%2C%092%2C%0927%2F2%2C%0D%0A0%2C%091%2C%09-3%2C%09-15%2C%0D%0A0%2C%090%2C%091%2C%098%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Eliminate the 3rd column\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C%0D%0A1%2C%090%2C%090%2C%09-5%2F2%2C%0D%0A0%2C%091%2C%090%2C%099%2C%0D%0A0%2C%090%2C%091%2C%098%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2.5 ALIGN=MIDDLE  ALT=\"x+=+-2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+9 ALIGN=MIDDLE  ALT=\"y+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+8 ALIGN=MIDDLE  ALT=\"z+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(iv) Cramer`s rule\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C4%2C%0D%0A2%2C%091%2C%091%2C%0912%2C%0D%0A6%2C%095%2C%09-3%2C%096%2C%0D%0A4%2C%09-1%2C%093%2C%095%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Write down the main matrix and find its determinant\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C3%2C%0D%0A2%2C%091%2C%091%2C%09%0D%0A6%2C%095%2C%09-3%2C%0D%0A4%2C%09-1%2C%093%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Δ = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A5%2A3%2B6%2A%28-1%29%2A1%2B4%2A1%28-3%29-%282%28-1%29%2A%28-3%29%29-%286%2A1%2A3%29-%284%2A5%2A1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Δ =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30-6-12+-6-18-20 ALIGN=MIDDLE  ALT=\"30-6-12+-6-18-20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Δ =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-32 ALIGN=MIDDLE  ALT=\"-32\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Replace the 1st column of the main matrix with the solution vector and find its determinant\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C3%2C12%2C%091%2C%091%2C%0D%0A6%2C%095%2C%09-3%2C%0D%0A5%2C%09-1%2C%093%29 ALIGN=MIDDLE  ALT=\"matrix%283%2C3%2C12%2C%091%2C%091%2C%0D%0A6%2C%095%2C%09-3%2C%0D%0A5%2C%09-1%2C%093%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Δ1 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=80 ALIGN=MIDDLE  ALT=\"80\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Replace the 2nd column of the main matrix with the solution vector and find its determinant\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C3%2C2%2C%0912%2C%091%2C%0D%0A6%2C%096%2C%09-3%2C%0D%0A4%2C%095%2C%093%29 ALIGN=MIDDLE  ALT=\"matrix%283%2C3%2C2%2C%0912%2C%091%2C%0D%0A6%2C%096%2C%09-3%2C%0D%0A4%2C%095%2C%093%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Δ2 =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-288 ALIGN=MIDDLE  ALT=\"+-288\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Replace the 3rd column of the main matrix with the solution vector and find its determinant\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%283%2C3%2C2%2C%091%2C%0912%2C%0D%0A6%2C%095%2C%096%2C%0D%0A4%2C%09-1%2C%095%29 ALIGN=MIDDLE  ALT=\"matrix%283%2C3%2C2%2C%091%2C%0912%2C%0D%0A6%2C%095%2C%096%2C%0D%0A4%2C%09-1%2C%095%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Δ3 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-256 ALIGN=MIDDLE  ALT=\"-256\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+ ALIGN=MIDDLE  ALT=\"x+\"   DISABLED_style= \"max-width:90%; height:auto;\" >= Δ1 / Δ = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=80+%2F+%28-32%29+=+-5%2F2 ALIGN=MIDDLE  ALT=\"80+%2F+%28-32%29+=+-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+ ALIGN=MIDDLE  ALT=\"y+\"   DISABLED_style= \"max-width:90%; height:auto;\" >= Δ2 / Δ = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-288%29+%2F+%28-32%29+=+9 ALIGN=MIDDLE  ALT=\"%28-288%29+%2F+%28-32%29+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" > = Δ3 / Δ =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28-256%29+%2F+%28-32%29+=+8 ALIGN=MIDDLE  ALT=\"+%28-256%29+%2F+%28-32%29+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "leaving to you to do:\r<BR>\n" );
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document.write( "(iii) Use Sarrus` rule to find determinant of the equation above\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );