document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209084>Question 1209084</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For a certain value of k, the system<BR>\n" );
document.write( "x + y + 3z = 10,<BR>\n" );
document.write( "4x + 5y + 6z = 7,<BR>\n" );
document.write( "kx - 3y + 2z = 3<BR>\n" );
document.write( "has no solutions. What is this value of k? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847654 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=847654\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>k = -16/9</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "The given system is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x+%2B+y+%2B+3z+=+10%2C4x+%2B+5y+%2B+6z+=+7%2Ckx+-+3y+%2B+2z+=+3%29 ALIGN=MIDDLE  ALT=\"system%28x+%2B+y+%2B+3z+=+10%2C4x+%2B+5y+%2B+6z+=+7%2Ckx+-+3y+%2B+2z+=+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "It converts to this matrix<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >1</TD><TD >3</TD><TD >10</TD></tr><tr><TD >4</TD><TD >5</TD><TD >6</TD><TD >7</TD></tr><tr><TD >k</TD><TD >-3</TD><TD >2</TD><TD >3</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "I placed the matrix in a table or grid format to separate out each entry. \r<BR>\n" );
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document.write( "From here we can apply row operations to get the matrix into Row Echelon Form (REF).<BR>\n" );
document.write( "This is where we turn each pivot into a 1, and zero out each item below the pivots. <BR>\n" );
document.write( "The pivots are along the northwest diagonal.\r<BR>\n" );
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document.write( "Here's what the steps look like.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >1</TD><TD >3</TD><TD >10</TD></tr><tr><TD >4</TD><TD >5</TD><TD >6</TD><TD >7</TD></tr><tr><TD >k</TD><TD >-3</TD><TD >2</TD><TD >3</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >1</TD><TD >3</TD><TD >10</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >-6</TD><TD >-33</TD><TD >R2 - 4*R1 --> R2</TD></tr><tr><TD >k</TD><TD >-3</TD><TD >2</TD><TD >3</TD><TD ></TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >1</TD><TD >3</TD><TD >10</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >-6</TD><TD >-33</TD><TD ></TD></tr><tr><TD >0</TD><TD >-3-k</TD><TD >2-3k</TD><TD >3-10k</TD><TD >R3 - k*R1 --> R3</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >1</TD><TD >3</TD><TD >10</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >-6</TD><TD >-33</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >-16-9k</TD><TD >-96-43k</TD><TD >R3 - (-3-k)*R2 --> R3</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >1</TD><TD >1</TD><TD >3</TD><TD >10</TD><TD ></TD></tr><tr><TD >0</TD><TD >1</TD><TD >-6</TD><TD >-33</TD><TD ></TD></tr><tr><TD >0</TD><TD >0</TD><TD >1</TD><TD >(-96-43k)/(-16-9k)</TD><TD >(1/(-16-9k))*R3 --> R3</TD></tr>\n" );
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document.write( "At this point we can stop since the pivots are 1 and each item below the pivot is 0.<BR>\n" );
document.write( "The third row leads to the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0x%2B0y%2B1z+=+%28-96-43k%29%2F%28-16-9k%29 ALIGN=MIDDLE  ALT=\"0x%2B0y%2B1z+=+%28-96-43k%29%2F%28-16-9k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-96-43k%29%2F%28-16-9k%29 ALIGN=MIDDLE  ALT=\"z+=+%28-96-43k%29%2F%28-16-9k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Set the denominator equal to zero and isolate k.<BR>\n" );
document.write( "-16-9k = 0<BR>\n" );
document.write( "-9k = 16<BR>\n" );
document.write( "<font color=red>k = -16/9</font>\r<BR>\n" );
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document.write( "If <font color=red>k = -16/9</font>, then the denominator of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-96-43k%29%2F%28-16-9k%29 ALIGN=MIDDLE  ALT=\"z+=+%28-96-43k%29%2F%28-16-9k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is zero, and it would mean there are no solutions to the original system.<BR>\n" );
document.write( "For any other value of k, that fraction is some real number; and we can use that value of z to find y, and then find x using back substitution.\r<BR>\n" );
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document.write( "Verification using <a  rel=nofollow HREF=\"https://www.wolframalpha.com/input?i=x+%2B+y+%2B+3z+%3D+10%2C4x+%2B+5y+%2B+6z+%3D+7%2C%28-16%2F9%29x+-+3y+%2B+2z+%3D+3\">WolframAlpha</a>\r<BR>\n" );
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document.write( "Somewhat similar questions are found <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Systems-of-equations/Systems-of-equations.faq.question.1206698.html\">here</a> and <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/college/linear/Linear_Algebra.faq.question.1204100.html\">here</a><BR>\n" );
document.write( "</font>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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