document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204100>Question 1204100</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the system of equations<BR>\n" );
document.write( "x + 2y - 3z = a<BR>\n" );
document.write( "2x + 6y - 11z = b<BR>\n" );
document.write( "x - 2y + 7z = c<BR>\n" );
document.write( "where a, b and c are three real numbers.<BR>\n" );
document.write( "1 What relation must the parameters a, b and c satisfy for the system of equations to have at least<BR>\n" );
document.write( "the system of equations has at least one solution?<BR>\n" );
document.write( "2. Assuming that a, b and c satisfy the relation that allows the system to have at least one solution, what relation must a, b and c satisfy?<BR>\n" );
document.write( "the system has at least one solution, calculate in function of a, b and c, the general solution of the system of equations using Gauss's method.<BR>\n" );
document.write( "Can the above linear system have a unique solution in R3? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840281 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=840281\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll convert the system of equations into a matrix as shown below.<BR>\n" );
document.write( "Then I'll get that matrix into row echelon form (REF)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Each matrix is presented as a table. In my opinion, the grid lines help separate things to make the entries look more cleaner. <BR>\n" );
document.write( "Normally however, matrix notation will not have these helpful grid lines.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >2</TD><TD >-3</TD><TD >a</TD></tr><tr><TD >2</TD><TD >6</TD><TD >-11</TD><TD >b</TD></tr><tr><TD >1</TD><TD >-2</TD><TD >7</TD><TD >c</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 >2</TD><TD >-3</TD><TD >a</TD><TD ></TD></tr><tr><TD >2</TD><TD >6</TD><TD >-11</TD><TD >b</TD><TD ></TD></tr><tr><TD >0</TD><TD >-4</TD><TD >10</TD><TD >c-a</TD><TD >R3  -  R1 --> R3</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "Notation like R3 - R1 --> R3 means we subtract rows 3 and 1, and store the results into row 3. <BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >2</TD><TD >-3</TD><TD >a</TD><TD ></TD></tr><tr><TD >0</TD><TD >2</TD><TD >-5</TD><TD >b-2a</TD><TD >R2  -  2*R1 --> R2</TD></tr><tr><TD >0</TD><TD >-4</TD><TD >10</TD><TD >c-a</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 >2</TD><TD >-3</TD><TD >a</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >-5/2</TD><TD >(b-2a)/2</TD><TD >(1/2)*R2 --> R2</TD></tr><tr><TD >0</TD><TD >-4</TD><TD >10</TD><TD >c-a</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 >2</TD><TD >-3</TD><TD >a</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >-5/2</TD><TD >(b-2a)/2</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >0</TD><TD >-5a+2b+c</TD><TD >R3 + 4*R2 --> R3</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "We have gone from the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C1%2C2%2C-3%2Ca%2C2%2C6%2C-11%2Cb%2C1%2C-2%2C7%2Cc%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C4%2C1%2C2%2C-3%2Ca%2C2%2C6%2C-11%2Cb%2C1%2C-2%2C7%2Cc%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C1%2C2%2C-3%2Ca%2C0%2C1%2C-5%2F2%2C%28b-2a%29%2F2%2C0%2C0%2C0%2C-5a%2B2b%2Bc%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The last row leads to the equation<BR>\n" );
document.write( "0x+0y+0z = -5a+2b+c<BR>\n" );
document.write( "or<BR>\n" );
document.write( "0 = -5a+2b+c\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Solve for c to get<BR>\n" );
document.write( "c = 5a-2b\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This will mean we have infinitely many solutions if and only if c = 5a-2b<BR>\n" );
document.write( "Otherwise, -5a+2b+c is nonzero and it causes a contradiction, and hence no solutions.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The values of 'a' and b can be any two real numbers you want.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's look at an example that has infinitely many solutions.<BR>\n" );
document.write( "a = 1, b = 2<BR>\n" );
document.write( "c = 5a-2b = 5*1-2*2 = 1<BR>\n" );
document.write( "Therefore this system<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2B2y-3z=1%2C2x%2B6y-11z=2%2Cx-2y%2B7z=1%29 ALIGN=MIDDLE  ALT=\"system%28x%2B2y-3z=1%2C2x%2B6y-11z=2%2Cx-2y%2B7z=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "is consistent and dependent.<BR>\n" );
document.write( "It has infinitely many solutions.<BR>\n" );
document.write( "I'll let the student verify this claim, and the later claims mentioned below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another example with infinitely many solutions.<BR>\n" );
document.write( "a = 5, b = 3<BR>\n" );
document.write( "c = 5a-2b = 5*5-2*3 = 19<BR>\n" );
document.write( "This system<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2B2y-3z=5%2C2x%2B6y-11z=3%2Cx-2y%2B7z=19%29 ALIGN=MIDDLE  ALT=\"system%28x%2B2y-3z=5%2C2x%2B6y-11z=3%2Cx-2y%2B7z=19%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "is consistent and dependent.<BR>\n" );
document.write( "It has infinitely many solutions.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's look at an example of a system that has no solutions.<BR>\n" );
document.write( "a = 1, b = 2, c = 3<BR>\n" );
document.write( "These a,b,c values do not satisfy the equation c = 5a-2b<BR>\n" );
document.write( "Therefore this system shown below has no solutions (it is inconsistent)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2B2y-3z=1%2C2x%2B6y-11z=2%2Cx-2y%2B7z=3%29 ALIGN=MIDDLE  ALT=\"system%28x%2B2y-3z=1%2C2x%2B6y-11z=2%2Cx-2y%2B7z=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As you can see, we cannot pick a trio of a,b,c values to have the system produce exactly one unique solution. <BR>\n" );
document.write( "Either we have infinitely many solutions, or none at all.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );