document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204569>Question 1204569</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Explain why, for any values of a, b, and c, the equations<BR>\n" );
document.write( "2x + 2y + 5z = a<BR>\n" );
document.write( "−3x + y − 2z = b<BR>\n" );
document.write( "x + z = c<BR>\n" );
document.write( "always have a unique solution.\r<BR>\n" );
document.write( "\n" );
document.write( "Find this unique solution (in terms of a, b, and c) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840898 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3821)</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=840898\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The given system<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x%2B2y%2B5z=a%2C-3x%2By-2z=b%2Cx%2Bz+=+c%29 ALIGN=MIDDLE  ALT=\"system%282x%2B2y%2B5z=a%2C-3x%2By-2z=b%2Cx%2Bz+=+c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "converts to this matrix<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C2%2C+2%2C+5%2C+a%2C+-3%2C+1%2C+-2%2C+b%2C+1%2C+0%2C+1%2C+c%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C2%2C+2%2C+5%2C+a%2C+-3%2C+1%2C+-2%2C+b%2C+1%2C+0%2C+1%2C+c%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Normally matrices do not have separating lines, but I think it's beneficial to have them. <BR>\n" );
document.write( "I'll place the items in a table like this<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >2</TD><TD >2</TD><TD >5</TD><TD >a</TD></tr><tr><TD >-3</TD><TD >1</TD><TD >-2</TD><TD >b</TD></tr><tr><TD >1</TD><TD >0</TD><TD >1</TD><TD >c</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's apply row operations to zero out the entries below each pivot.\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >1</TD><TD >5/2</TD><TD >a/2</TD><TD >(1/2)*R1 --> R1</TD></tr><tr><TD >-3</TD><TD >1</TD><TD >-2</TD><TD >b</TD><TD ></TD></tr><tr><TD >1</TD><TD >0</TD><TD >1</TD><TD >c</TD><TD ></TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Notation like (1/2)*R1 --> R1 means we take half of each element in row 1 (aka R1). Then store the results in R1.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >1</TD><TD >5/2</TD><TD >a/2</TD><TD ></TD></tr><tr><TD >0</TD><TD >4</TD><TD >11/2</TD><TD >3a/2+b</TD><TD >R2 + 3*R1 --> R2</TD></tr><tr><TD >1</TD><TD >0</TD><TD >1</TD><TD >c</TD><TD ></TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Something like R2 + 3*R1 --> R2 will mean that we triple everything in R1, then add those results to R2. Store the results in R2.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >1</TD><TD >5/2</TD><TD >a/2</TD><TD ></TD></tr><tr><TD >0</TD><TD >4</TD><TD >11/2</TD><TD >3a/2+b</TD><TD ></TD></tr><tr><TD >0</TD><TD >-1</TD><TD >-3/2</TD><TD >c  -  a/2</TD><TD >R3  -  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 >1</TD><TD >5/2</TD><TD >a/2</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >11/8</TD><TD >3a/8+b/4</TD><TD >(1/4)*R2 --> R2</TD></tr><tr><TD >0</TD><TD >-1</TD><TD >-3/2</TD><TD >c  -  a/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 >1</TD><TD >5/2</TD><TD >a/2</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >11/8</TD><TD >3a/8+b/4</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >-1/8</TD><TD >-a/8 + b/4 + c</TD><TD >R3 + R2 --> 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 >1</TD><TD >5/2</TD><TD >a/2</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >11/8</TD><TD >3a/8+b/4</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >a-2b-8c</TD><TD >-8*R3 --> R3</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "At this point the matrix is in row echelon form (REF), but we haven't reached RREF just yet. <BR>\n" );
document.write( "Let's keep row reducing until all of the non-pivot entries are turned to 0.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >1</TD><TD >5/2</TD><TD >a/2</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >0</TD><TD >-a+3b+11c</TD><TD >R2  -  (11/8)*R3 --> R2</TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >a-2b-8c</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 >1</TD><TD >0</TD><TD >-2a+5b+20c</TD><TD >R1  -  (5/2)*R3 --> R3</TD></tr><tr><TD >0</TD><TD >1</TD><TD >0</TD><TD >-a+3b+11c</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >a-2b-8c</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 >0</TD><TD >-a+2b+9c</TD><TD >R1  -  R2 --> R1</TD></tr><tr><TD >0</TD><TD >1</TD><TD >0</TD><TD >-a+3b+11c</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >a-2b-8c</TD><TD ></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "The matrix is now in Reduced Row Echelon Form (RREF).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The system has <font color=red>exactly one solution</font>, and it is when:<BR>\n" );
document.write( "<font color=red size=4>x = -a+2b+9c<BR>\n" );
document.write( "y = -a+3b+11c<BR>\n" );
document.write( "z = a-2b-8c</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Confirmation using WolframAlpha<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.wolframalpha.com/input/?i=2x%2B2y%2B5z%3Da%2C-3x%2By-2z%3Db%2Cx%2Bz%3Dc\">https://www.wolframalpha.com/input/?i=2x%2B2y%2B5z%3Da%2C-3x%2By-2z%3Db%2Cx%2Bz%3Dc</a><BR>\n" );
document.write( "The search input to type in is \"<font color=blue>2x+2y+5z=a,-3x+y-2z=b,x+z=c</font>\" without quotes.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The CAS feature in GeoGebra can also be used.<BR>\n" );
document.write( "There are two options to input<BR>\n" );
document.write( "Either <BR>\n" );
document.write( "<font color=blue>Solve[{2x+2y+5z=a,-3x+y-2z=b,x+z=c}]</font><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<font color=blue>ReducedRowEchelonForm[{{2, 2, 5, a}, {-3, 1, -2, b}, {1, 0, 1, c}}]</font><BR>\n" );
document.write( "The way GeoGebra handles matrices is that they are a collection of lists. <BR>\n" );
document.write( "Each list is enclosed in curly braces, which represents a particular row.<BR>\n" );
document.write( "There are likely other CAS based matrix calculators out there that can offer similar features.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Some more practice with matrix row reduction<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/college/linear/Linear_Algebra.faq.question.1204100.html\">https://www.algebra.com/algebra/college/linear/Linear_Algebra.faq.question.1204100.html</a><BR>\n" );
document.write( "and <BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Matrices-and-determiminant.faq.question.1203997.html\">https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Matrices-and-determiminant.faq.question.1203997.html</a><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );