document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=120653>Question 120653</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Business and finance. In planning for a new item, a manufacturer assumes that<BR>\n" );
document.write( "the number of items produced x and the cost in dollars C of producing these items<BR>\n" );
document.write( "are related by a linear equation. Projections are that 100 items will cost $10,000 to<BR>\n" );
document.write( "produce and that 300 items will cost $22,000 to produce. Find the equation that<BR>\n" );
document.write( "relates C and x. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #88441 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=88441\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x=items, y=cost to produce x amount of items\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since 100 items will cost $10,000, this means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=100 ALIGN=MIDDLE  ALT=\"x=100\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10000 ALIGN=MIDDLE  ALT=\"y=10000\"   DISABLED_style= \"max-width:90%; height:auto;\" > which results in the point (100,10000)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Also since 300 items will cost $22,000, this means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=300 ALIGN=MIDDLE  ALT=\"x=300\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=22000 ALIGN=MIDDLE  ALT=\"y=22000\"   DISABLED_style= \"max-width:90%; height:auto;\" > which results in the point (300,22000)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we have the two points: (100,10000) and (300,22000)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's find the equation of the line through these points\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First lets find the slope through the points (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10000 ALIGN=MIDDLE  ALT=\"10000\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=300 ALIGN=MIDDLE  ALT=\"300\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22000 ALIGN=MIDDLE  ALT=\"22000\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the slope formula (note: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_{1},y_{1}\right)\"> is the first point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10000 ALIGN=MIDDLE  ALT=\"10000\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_{2},y_{2}\right)\"> is the second point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=300 ALIGN=MIDDLE  ALT=\"300\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22000 ALIGN=MIDDLE  ALT=\"22000\"   DISABLED_style= \"max-width:90%; height:auto;\" >))\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%2822000-10000%29%2F%28300-100%29 ALIGN=MIDDLE  ALT=\"m=%2822000-10000%29%2F%28300-100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=22000 ALIGN=MIDDLE  ALT=\"y%5B2%5D=22000\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=10000 ALIGN=MIDDLE  ALT=\"y%5B1%5D=10000\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=300 ALIGN=MIDDLE  ALT=\"x%5B2%5D=300\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=100 ALIGN=MIDDLE  ALT=\"x%5B1%5D=100\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (these are the coordinates of given points)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=+12000%2F200 ALIGN=MIDDLE  ALT=\"m=+12000%2F200\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22000-10000 ALIGN=MIDDLE  ALT=\"22000-10000\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12000 ALIGN=MIDDLE  ALT=\"12000\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Subtract the terms in the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=300-100 ALIGN=MIDDLE  ALT=\"300-100\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200 ALIGN=MIDDLE  ALT=\"200\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=60 ALIGN=MIDDLE  ALT=\"m=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
document.write( "\n" );
document.write( "  \r<BR>\n" );
document.write( "\n" );
document.write( "So the slope is\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=60 ALIGN=MIDDLE  ALT=\"m=60\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's use the point-slope formula to find the equation of the line:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------Point-Slope Formula------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B1%5D=m%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the slope, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\textrm{x_{1},y_{1}}\right)\"> is one of the given points\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So lets use the Point-Slope Formula to find the equation of the line\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-10000=%2860%29%28x-100%29 ALIGN=MIDDLE  ALT=\"y-10000=%2860%29%28x-100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=60 ALIGN=MIDDLE  ALT=\"m=60\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=100 ALIGN=MIDDLE  ALT=\"x%5B1%5D=100\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=10000 ALIGN=MIDDLE  ALT=\"y%5B1%5D=10000\"   DISABLED_style= \"max-width:90%; height:auto;\" > (these values are given)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-10000=60x%2B%2860%29%28-100%29 ALIGN=MIDDLE  ALT=\"y-10000=60x%2B%2860%29%28-100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-10000=60x-6000 ALIGN=MIDDLE  ALT=\"y-10000=60x-6000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60 ALIGN=MIDDLE  ALT=\"60\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-100 ALIGN=MIDDLE  ALT=\"-100\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6000 ALIGN=MIDDLE  ALT=\"-6000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=60x-6000%2B10000 ALIGN=MIDDLE  ALT=\"y=60x-6000%2B10000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10000 ALIGN=MIDDLE  ALT=\"10000\"   DISABLED_style= \"max-width:90%; height:auto;\" > to  both sides to isolate y\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=60x%2B4000 ALIGN=MIDDLE  ALT=\"y=60x%2B4000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6000 ALIGN=MIDDLE  ALT=\"-6000\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10000 ALIGN=MIDDLE  ALT=\"10000\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4000 ALIGN=MIDDLE  ALT=\"4000\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the equation of the line which goes through the points (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10000 ALIGN=MIDDLE  ALT=\"10000\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=300 ALIGN=MIDDLE  ALT=\"300\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22000 ALIGN=MIDDLE  ALT=\"22000\"   DISABLED_style= \"max-width:90%; height:auto;\" >)  is:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=60x%2B4000 ALIGN=MIDDLE  ALT=\"y=60x%2B4000\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now simply replace y with C(x) to get the equation \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28x%29=60x%2B4000 ALIGN=MIDDLE  ALT=\"C%28x%29=60x%2B4000\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: C is much easier to remember as the <b>C</b>ost. Also notice how the equation is now in function notation)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );