document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=183071>Question 183071</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0471163627/algebrahomeworkh>Contemporary Linear Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Let u = (1,-1,3,5) and v = (2,1,0,-3). Find scalars a and b so that <BR>\n" );
document.write( "au + bv = (1,-4,9,18).\r<BR>\n" );
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document.write( "I would like to know the steps on how to solve for scalars a and b. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #137442 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=137442\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> au+bv = (1,-4,9,18) ... Start with the given equation\r<BR>\n" );
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document.write( "a(1,-1,3,5)+b(2,1,0,-3) = (1,-4,9,18) ... Plug in the given vectors\r<BR>\n" );
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document.write( "(a,-a,3a,5a)+(2b,b,0,-3b) = (1,-4,9,18) ... Multiply the scalars by EVERY element in the vectors\r<BR>\n" );
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document.write( "(a+2b,-a+b,3a,5a-3b) = (1,-4,9,18) ... Add the vectors by adding the corresponding components.\r<BR>\n" );
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document.write( "Now because both vectors on both sides are equal in dimension, this tells us that the left components correspond to the right components. This means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B2b=1 ALIGN=MIDDLE  ALT=\"a%2B2b=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a%2Bb=-4 ALIGN=MIDDLE  ALT=\"-a%2Bb=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a=9 ALIGN=MIDDLE  ALT=\"3a=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5a-3b=18 ALIGN=MIDDLE  ALT=\"5a-3b=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Which gives us the system of equations:\r<BR>\n" );
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document.write( "We can use any method to solve this system (substitution, elimination, matrix, etc), but notice how the third equation is composed of only one variable in which we can easily solve for.\r<BR>\n" );
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document.write( "Let's now solve the third equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a=9 ALIGN=MIDDLE  ALT=\"3a=9\"   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=3a=9 ALIGN=MIDDLE  ALT=\"3a=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the third equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=9%2F3 ALIGN=MIDDLE  ALT=\"a=9%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 3 to isolate \"a\".\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B2b=1 ALIGN=MIDDLE  ALT=\"a%2B2b=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Go back to the first equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B2b=1 ALIGN=MIDDLE  ALT=\"3%2B2b=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   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=2b=1-3 ALIGN=MIDDLE  ALT=\"2b=1-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b=-2 ALIGN=MIDDLE  ALT=\"2b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=%28-2%29%2F%282%29 ALIGN=MIDDLE  ALT=\"b=%28-2%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   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=b=-1 ALIGN=MIDDLE  ALT=\"b=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the scalars are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-1 ALIGN=MIDDLE  ALT=\"b=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In other words, \r<BR>\n" );
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document.write( "3(1,-1,3,5)-1*(2,1,0,-3) = (1,-4,9,18) </SPAN>\n" );
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