document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204117>Question 1204117</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Lauren is planning a catered dinner party for her parents with a budget of 396.00. She has selected two options: a chicken dinner that costs 9 dollars a plate and a steak dinner that cost 12 dollars a plate. Lauren is working on the guests list and must also determine how many of each meal to order. Set up an equation and solve.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840259 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=840259\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The problem is poorly written.  In real life, if the budget is $396, it would mean that the MAXIMUM you could spend is $396.  In that case, solving the problem would involve an inequality rather than an equation.  However, since the instructions say to write and solve an equation, it appears that we are supposed to spend the whole $396.<br><BR>\n" );
document.write( "But then there are many combinations of chicken and steak dinners that will cost a total of $396, so we can't \"solve\" the problem in the sense of finding \"the\" answer.<br><BR>\n" );
document.write( "So to work the problem we need to (1) assume that we are spending the whole $396 and (2) we are looking for the solution set and not \"the\" solution.<br><BR>\n" );
document.write( "Let C be the number of chicken dinners and S be the number of steak dinners.  Then the total cost of $396 is $9 for each chicken dinner plus $12 for each steak dinner:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9C%2B12S=396 ALIGN=MIDDLE  ALT=\"9C%2B12S=396\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3C%2B4S=132 ALIGN=MIDDLE  ALT=\"3C%2B4S=132\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is a linear Diophantine equation -- there are two unknowns and only a single equation, with the number of solutions being limited by the fact that the unknowns have whole number values.<br><BR>\n" );
document.write( "There are formal methods for finding the set of solutions; but in this case perhaps a more informal method using logical reasoning is appropriate.<br><BR>\n" );
document.write( "The maximum number of chicken dinners is if there are no steak dinners:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3C%2B4%280%29=132 ALIGN=MIDDLE  ALT=\"3C%2B4%280%29=132\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3C=132 ALIGN=MIDDLE  ALT=\"3C=132\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=44 ALIGN=MIDDLE  ALT=\"C=44\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The maximum number of steak dinners is if there are no chicken dinners:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%280%29%2B4S=132 ALIGN=MIDDLE  ALT=\"3%280%29%2B4S=132\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4S=132 ALIGN=MIDDLE  ALT=\"4S=132\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=33 ALIGN=MIDDLE  ALT=\"S=33\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So two solutions to the problem are 44 chicken dinners and 0 steak dinners, and 0 chicken dinners and 33 steak dinners.<br><BR>\n" );
document.write( "We can find the other solutions by seeing that the cost of 4 chicken dinners is the same as the cost of 3 steak dinners.  So, given any solution that spends the whole $396, we can find another solution by increasing the number of chicken dinners by 4 and decreasing the number of steak dinners, or the other way around.<br><BR>\n" );
document.write( "Doing that, we get the following solution set of values (C,S):<br><BR>\n" );
document.write( "(44,0)<BR>\n" );
document.write( "(40,3)<BR>\n" );
document.write( "(36,6)<BR>\n" );
document.write( "(32,9)<BR>\n" );
document.write( "(28,12)<BR>\n" );
document.write( "(24,15)<BR>\n" );
document.write( "(20,18)<BR>\n" );
document.write( "(16,21)<BR>\n" );
document.write( "(12,24)<BR>\n" );
document.write( "(8,27)<BR>\n" );
document.write( "(4,30)<BR>\n" );
document.write( "(0,33)<br><BR>\n" );
document.write( "That's 12 possible combinations of chicken and steak dinner that will spend the entire $396 budget.<br><BR>\n" );
document.write( "To get \"AN\" answer to the problem, we would need to have another piece of information.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );