document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207192>Question 1207192</A>: <I><SPAN STYLE=\"word-break: break-all;\"> given the system of equations 2x-3y-9z=20 x+3z=-2 -3x+y-4z=-2 find the complete solution write x and y as functions of z  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844930 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=844930\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x-3y-9z=20%2Cx%2B3z=-2%2C+-3x%2By-4z=-2%29 ALIGN=MIDDLE  ALT=\"system%282x-3y-9z=20%2Cx%2B3z=-2%2C+-3x%2By-4z=-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is equivalent to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x-3y-9z=20%2C1x%2B0y%2B3z=-2%2C+-3x%2B1y-4z=-2%29 ALIGN=MIDDLE  ALT=\"system%282x-3y-9z=20%2C1x%2B0y%2B3z=-2%2C+-3x%2B1y-4z=-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "That system converts to this augmented matrix.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >2</TD><TD >-3</TD><TD >-9</TD><TD >20</TD></tr><tr><TD >1</TD><TD >0</TD><TD >3</TD><TD >-2</TD></tr><tr><TD >-3</TD><TD >1</TD><TD >-4</TD><TD >-2</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Normally the grid lines aren't present to separate each item. But I decided to make it into a table format.\r<BR>\n" );
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document.write( "Let's apply Gauss Jordan Elimination to get the matrix into Reduced Row Echelon Form (RREF).<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >2</TD><TD >-3</TD><TD >-9</TD><TD >20</TD></tr><tr><TD >1</TD><TD >0</TD><TD >3</TD><TD >-2</TD></tr><tr><TD >-3</TD><TD >1</TD><TD >-4</TD><TD >-2</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >0</TD><TD >3</TD><TD >-2</TD><TD >R1 <--> R2</TD></tr><tr><TD >2</TD><TD >-3</TD><TD >-9</TD><TD >20</TD><TD ></TD></tr><tr><TD >-3</TD><TD >1</TD><TD >-4</TD><TD >-2</TD><TD ></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >0</TD><TD >3</TD><TD >-2</TD><TD ></TD></tr><tr><TD >0</TD><TD >-3</TD><TD >-15</TD><TD >24</TD><TD >R2 - 2*R1 --> R2</TD></tr><tr><TD >-3</TD><TD >1</TD><TD >-4</TD><TD >-2</TD><TD ></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >0</TD><TD >3</TD><TD >-2</TD><TD ></TD></tr><tr><TD >0</TD><TD >-3</TD><TD >-15</TD><TD >24</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >5</TD><TD >-8</TD><TD >R3 + 3*R1 --> R3</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >0</TD><TD >3</TD><TD >-2</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >5</TD><TD >-8</TD><TD >R2 <--> R3</TD></tr><tr><TD >0</TD><TD >-3</TD><TD >-15</TD><TD >24</TD><TD ></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >0</TD><TD >3</TD><TD >-2</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >5</TD><TD >-8</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >R3 + 3*R2 --> R3</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "Here is a step-by-step calculator that is very useful to row reduce matrices<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"http://www.math.odu.edu/~bogacki/lat/\">http://www.math.odu.edu/~bogacki/lat/</a><BR>\n" );
document.write( "It is called \"linear algebra toolkit\".<BR>\n" );
document.write( "Click the \"Enter\" link and then go to \"Row operation calculator\". Let me know if you have any questions about this calculator. \r<BR>\n" );
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document.write( "More practice with gauss-jordan elimination<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/coordinate/Linear-systems.faq.question.1203611.html\">https://www.algebra.com/algebra/homework/coordinate/Linear-systems.faq.question.1203611.html</a>\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "To briefly summarize, we have gone from this matrix <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C2%2C-3%2C-9%2C20%2C1%2C0%2C3%2C-2%2C-3%2C1%2C-4%2C-2%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C2%2C-3%2C-9%2C20%2C1%2C0%2C3%2C-2%2C-3%2C1%2C-4%2C-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to this matrix<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C1%2C0%2C3%2C-2%2C0%2C1%2C5%2C-8%2C0%2C0%2C0%2C0%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C1%2C0%2C3%2C-2%2C0%2C1%2C5%2C-8%2C0%2C0%2C0%2C0%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The row of all zeros tells us that we will have infinitely many solutions. This system is consistent and dependent.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "The 2nd matrix converts back to this system<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2B3z=-2%2Cy%2B5z=-8%29 ALIGN=MIDDLE  ALT=\"system%28x%2B3z=-2%2Cy%2B5z=-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and this is what results when we get each z term to the other side<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=-3z-2%2Cy=-5z-8%29 ALIGN=MIDDLE  ALT=\"system%28x=-3z-2%2Cy=-5z-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Therefore each of the infinitely many solutions are of the form (x,y,z) = (-3z-2,-5z-8,z) where z is any real number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Examples:<BR>\n" );
document.write( "If z = 0 then (x,y,z) = (-3z-2,-5z-8,z)= (-3*0-2,-5*0-8,0) = (-2,-8,0) is a solution<BR>\n" );
document.write( "If z = 1 then (x,y,z) = (-3z-2,-5z-8,z)= (-3*1-2,-5*1-8,1) = (-5,-13,1) is a solution<BR>\n" );
document.write( "If z = 2 then (x,y,z) = (-3z-2,-5z-8,z)= (-3*2-2,-5*2-8,2) = (-8,-18,2) is a solution<BR>\n" );
document.write( "If z = 3 then (x,y,z) = (-3z-2,-5z-8,z)= (-3*3-2,-5*3-8,3) = (-11,-23,3) is a solution<BR>\n" );
document.write( "All of these solution points are located on the same straight line. <BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );