document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=708648>Question 708648</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A plant food is to be made from three chemicals. The combined amounts of the first two chemicals must be 80% of the resulting mix. The second and third chemical must be in the ratio 4 to 3 by weight. How much of each chemical is needed to make 700 kg of the plant food? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #436198 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39618)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=436198\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Use a, b, and c as the quantities each chemical.<BR>\n" );
document.write( "These initial equations can be made from the problem description:\r<BR>\n" );
document.write( "\n" );
document.write( "The total was to be 700 pounds, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc=700 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc=700\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The 80% requirement gives, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%2F700=0.8 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%2F700=0.8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The ratio information indicates, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fc=4%2F3 ALIGN=MIDDLE  ALT=\"b%2Fc=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Rework the last two equations into these:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb=560 ALIGN=MIDDLE  ALT=\"a%2Bb=560\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b-%284%2F3%29c=0 ALIGN=MIDDLE  ALT=\"b-%284%2F3%29c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The system of equations to work with then is this one:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc=700 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc=700\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb=560 ALIGN=MIDDLE  ALT=\"a%2Bb=560\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b-%284%2F3%29c=0 ALIGN=MIDDLE  ALT=\"b-%284%2F3%29c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Matrix operations is the way to continue here, at least partly.<BR>\n" );
document.write( "That system can give us this matrix:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C4%2C1%2C1%2C1%2C700%2C%0D%0A1%2C1%2C0%2C560%2C%0D%0A0%2C1%2C-4%2F3%2C0%0D%0A%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I briefly list the row operations but not here show the work.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5B1%5D-R%5B2%5D=+R%5B2%5D ALIGN=MIDDLE  ALT=\"R%5B1%5D-R%5B2%5D=+R%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, immediately giving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=140 ALIGN=MIDDLE  ALT=\"c=140\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5B3%5D%2B%284%2F3%29R%5B1%5D=R%5B3%5D ALIGN=MIDDLE  ALT=\"R%5B3%5D%2B%284%2F3%29R%5B1%5D=R%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, giving b=186 2/3, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=186%262%2F3 ALIGN=MIDDLE  ALT=\"b=186%262%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now having b and c, the 700 summation equation can be used and b and c simply subsituted to find a (not continuing further with matrix operations).  <BR>\n" );
document.write( "result for a:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=373%262%2F3 ALIGN=MIDDLE  ALT=\"a=373%262%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=373%261%2F3 ALIGN=MIDDLE  ALT=\"a=373%261%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=186%262%2F3 ALIGN=MIDDLE  ALT=\"b=186%262%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=140 ALIGN=MIDDLE  ALT=\"c=140\"   DISABLED_style= \"max-width:90%; height:auto;\" >, all in pounds. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );