document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=629335>Question 629335</A>: <I><SPAN STYLE=\"word-break: break-all;\">                                     <BR>\n" );
document.write( "Express the vector v= [-17]  as a linear combination of x = [-2] and y = [-5]<BR>\n" );
document.write( "                                  [ 50 ]                                                 [ 5 ]              [ -6]\r<BR>\n" );
document.write( "\n" );
document.write( "v=__x + __ y. \r<BR>\n" );
document.write( "\n" );
document.write( "How could I express this? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #396209 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=396209\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm assuming that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v+=+%28matrix%282%2C+1%2C+-17%2C+50%29%29 ALIGN=MIDDLE  ALT=\"v+=+%28matrix%282%2C+1%2C+-17%2C+50%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28matrix%282%2C+1%2C+-2%2C+5%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28matrix%282%2C+1%2C+-2%2C+5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28matrix%282%2C+1%2C+-5%2C+-6%29%29 ALIGN=MIDDLE  ALT=\"y+=+%28matrix%282%2C+1%2C+-5%2C+-6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "The problem is to find what number times x plus what number times y adds up to v. Let's call these two numbers a and b. So we're looking for what a and b make<BR>\n" );
document.write( "v = a*x + b*y<BR>\n" );
document.write( "true.<br><BR>\n" );
document.write( "Substituting the vectors/column matrices in for v, x and y out equation is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C+1%2C+-17%2C+50%29%29+=+a%2A%28matrix%282%2C+1%2C+-2%2C+5%29%29+%2B+b%2A%28matrix%282%2C+1%2C+-5%2C+-6%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Performing the scalar multiplication on the right side we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C+1%2C+-17%2C+50%29%29+=+%28matrix%282%2C+1%2C+-2a%2C+5a%29%29+%2B+%28matrix%282%2C+1%2C+-5b%2C+-6b%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding the vectors/column matrices on the right we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C+1%2C+-17%2C+50%29%29+=+%28matrix%282%2C+1%2C+-2a+%2B+%28-5b%29%2C+5a+%2B+%28-6b%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By the definition of matrix/vector equality, these are equal only if each element of one is equal to the corresponding element of the other. So<BR>\n" );
document.write( "-17 = -2a + (-5b) and<BR>\n" );
document.write( "50 = 5a + (-6b)<br><BR>\n" );
document.write( "To find the correct values for a and b we solve this system. You have probably learned several ways to solve a system of linear equations like this: Substitution, Linear Combination/Elimination, several matrix methods, determinants/Cramer's Rule, etc. I'm going to use Linear Combination (but any of them will work and give the right answers).<BR>\n" );
document.write( "Multiplying the first equation by 5 and the second equation by 2 we get:<BR>\n" );
document.write( "-85 = -10a + (-25b)<BR>\n" );
document.write( "100 = 10a + (-12b)<BR>\n" );
document.write( "Adding the two equations together we get:<BR>\n" );
document.write( "15 = -37b<BR>\n" );
document.write( "Dividing both sides by -37 we get:<BR>\n" );
document.write( "-15/37 = b<BR>\n" );
document.write( "Now we use this value and one of the earlier equations to find a:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-17+=+-2a+%2B+%28-5%29%28-15%2F37%29 ALIGN=MIDDLE  ALT=\"-17+=+-2a+%2B+%28-5%29%28-15%2F37%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-17+=+-2a+%2B+75%2F37 ALIGN=MIDDLE  ALT=\"-17+=+-2a+%2B+75%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtracting 75/37 to each side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-17+-+75%2F37+=+-2a+ ALIGN=MIDDLE  ALT=\"-17+-+75%2F37+=+-2a+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-629%2F37+-+75%2F37+=+-2a ALIGN=MIDDLE  ALT=\"-629%2F37+-+75%2F37+=+-2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-704%2F37+=+-2a ALIGN=MIDDLE  ALT=\"-704%2F37+=+-2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing by -2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=352%2F37+=+a ALIGN=MIDDLE  ALT=\"352%2F37+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v+=+%28352%2F37%29x+%2B+%28-15%2F37%29y ALIGN=MIDDLE  ALT=\"v+=+%28352%2F37%29x+%2B+%28-15%2F37%29y\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );