document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203103>Question 1203103</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A customer bought 2 fruit cups and 3 sandwiches for US$ 19.\r<BR>\n" );
document.write( "\n" );
document.write( "When she bought 6 fruit cups and 8 sandwiches her bill was US $54.\r<BR>\n" );
document.write( "\n" );
document.write( "•	Write two equations to represent the information given above.<BR>\n" );
document.write( "•	Calculate the cost for ONE fruit cup.<BR>\n" );
document.write( "•	Hence, determine the cost per sandwich.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your friend, Manuel, got this question on a take home assignment. Having solved the pair of equations, he believes that the cost per fruit cup is US$3 and the cost per sandwich is US $5.\r<BR>\n" );
document.write( "\n" );
document.write( "a)	Write the pair of equations, clearly defining each variable used.  			[1 mark]\r<BR>\n" );
document.write( "\n" );
document.write( "b)	Use a graphical method to tell whether Manuel is correct. You must clearly label the solution and explicitly state your position on Manuel’s answer. 				           [2 marks]\r<BR>\n" );
document.write( "\n" );
document.write( "c)	Show Manuel an alternative approach on your College Algebra syllabus for solving pairs of simultaneous equations. You should name the method you will use and show full working.													[3 marks]\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838392 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=838392\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = cost of 1 fruit cup<BR>\n" );
document.write( "y = cost of 1 sandwich<BR>\n" );
document.write( "Each cost is in dollars.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x = cost of 2 fruit cups<BR>\n" );
document.write( "3y = cost of 3 sandwiches<BR>\n" );
document.write( "2x+3y = cost of 2 fruit cups and 3 sandwiches<BR>\n" );
document.write( "2x+3y = 19\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Through a similar thought process, the other equation would be 6x+8y = 54\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The system of equations is<BR>\n" );
document.write( "2x+3y = 19<BR>\n" );
document.write( "6x+8y = 54\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "======================================================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (b)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use GeoGebra, Desmos, TI83/84, or any graphing tool to plot the equations we found in the previous part.<BR>\n" );
document.write( "Another approach is to plot the equations by hand on a piece of graph paper.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2x+3y = 19 goes through the points (2,5) and (5,3)<BR>\n" );
document.write( "6x+8y = 54 goes through the points (1,6) and (5,3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The two lines intersect at the point (5,3)<BR>\n" );
document.write( "It is the single solution to the system.<BR>\n" );
document.write( "This breaks down to x = 5 and y = 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, 1 fruit cup costs $5 and 1 sandwich costs $3<BR>\n" );
document.write( "Manuel has the two values incorrectly swapped.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "======================================================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (c)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can use the elimination method as one approach.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Triple each side of the 1st equation<BR>\n" );
document.write( "2x+3y = 19<BR>\n" );
document.write( "3(2x+3y) = 3*19<BR>\n" );
document.write( "6x+9y = 57\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This original system<BR>\n" );
document.write( "2x+3y = 19<BR>\n" );
document.write( "6x+8y = 54\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "is equivalent to this system<BR>\n" );
document.write( "6x+9y = 57<BR>\n" );
document.write( "6x+8y = 54\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From here, subtract straight down.<BR>\n" );
document.write( "The x terms cancel out because 6x-6x = 0x = 0<BR>\n" );
document.write( "The y terms turn into 9y-8y = 1y = y<BR>\n" );
document.write( "The right-hand-sides become 57-54 = 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Ultimately this boils down to y = 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use this y value to find x.<BR>\n" );
document.write( "2x+3y = 19<BR>\n" );
document.write( "2x+3*3 = 19<BR>\n" );
document.write( "2x+9 = 19<BR>\n" );
document.write( "2x = 19-9<BR>\n" );
document.write( "2x = 10<BR>\n" );
document.write( "x = 10/2<BR>\n" );
document.write( "x = 5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Or we could pick on the other equation<BR>\n" );
document.write( "6x+8y = 54<BR>\n" );
document.write( "6x+8*3 = 54<BR>\n" );
document.write( "6x+24 = 54<BR>\n" );
document.write( "6x = 54-24<BR>\n" );
document.write( "6x = 30<BR>\n" );
document.write( "x = 30/6<BR>\n" );
document.write( "x = 5<BR>\n" );
document.write( "Either way, we have determined that x = 5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So that's how we can back up part (b) when we mentioned x = 5 and y = 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check:<BR>\n" );
document.write( "Plug x = 5 and y = 3 into the 1st equation<BR>\n" );
document.write( "2y+3y = 19<BR>\n" );
document.write( "2*5+3*3 = 19<BR>\n" );
document.write( "10+9 = 19<BR>\n" );
document.write( "19 = 19<BR>\n" );
document.write( "Repeat for the other equation<BR>\n" );
document.write( "6x+8y = 54<BR>\n" );
document.write( "6*5+8*3 = 54<BR>\n" );
document.write( "30+24 = 54<BR>\n" );
document.write( "54 = 54<BR>\n" );
document.write( "Both equations are true for this pair of x,y values. The answer is fully confirmed.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );