document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202814>Question 1202814</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The Unique Gifts catalog lists a \"super loud and vibrating alarm clock.\" Their records indicate the following information on the relation of monthly supply and demand quantities to the price of the clock.\r<BR>\n" );
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document.write( "Demand    	Supply    	Price<BR>\n" );
document.write( "166     	131     	$34<BR>\n" );
document.write( "146     	181     	$42\r<BR>\n" );
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document.write( "Use this information to find the following.\r<BR>\n" );
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document.write( "(a) points on the demand linear equation\r<BR>\n" );
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document.write( "(x, p) =(	) (smaller x-value)<BR>\n" );
document.write( "(x, p) =(	) (larger x-value)\r<BR>\n" );
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document.write( "points on the supply linear equation<BR>\n" );
document.write( "(x, p) =(	)(smaller x-value)<BR>\n" );
document.write( "(x, p) =(	)(larger x-value)\r<BR>\n" );
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document.write( "(b) the demand equation p<BR>\n" );
document.write( "p = \r<BR>\n" );
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document.write( "(c) the supply equation p<BR>\n" );
document.write( "p = <BR>\n" );
document.write( " \r<BR>\n" );
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document.write( "(d) the equilibrium quantity and price<BR>\n" );
document.write( "Equilibrium occurs when the price of the clock is $________ and the quantity is _________. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838525 by </B><A HREF=/tutors/aboutme.mpl?userid=jekoishun><B>jekoishun(6)</B></A><img src=\"/images/stars/stars-05.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jekoishun><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=838525\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You can find the equations for both demand and supply using the information provided. First, we need to find the slope and the y-intercept for both the demand and the supply lines.\r<BR>\n" );
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document.write( "(a) The points on the demand linear equation are given by the pairs of quantity demanded and price:<BR>\n" );
document.write( "\( (x, p) = (166, 34) \) (smaller x-value)<BR>\n" );
document.write( "\( (x, p) = (146, 42) \) (larger x-value)\r<BR>\n" );
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document.write( "The points on the supply linear equation are given by the pairs of quantity supplied and price:<BR>\n" );
document.write( "\( (x, p) = (131, 34) \) (smaller x-value)<BR>\n" );
document.write( "\( (x, p) = (181, 42) \) (larger x-value)\r<BR>\n" );
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document.write( "(b) To find the demand equation, we will use the two points on the demand line:<BR>\n" );
document.write( "\( (166, 34) \) and \( (146, 42) \)<BR>\n" );
document.write( "The slope of the demand line can be found using:<BR>\n" );
document.write( "\[ m = \frac{{42 - 34}}{{146 - 166}} = \frac{{8}}{{-20}} = -\frac{2}{5} \]<BR>\n" );
document.write( "We can use one of the points to find the y-intercept b:<BR>\n" );
document.write( "\[ 34 = -\frac{2}{5} \cdot 166 + b \]<BR>\n" );
document.write( "\[ b = \frac{2 \cdot 166}{5} + 34 = \frac{332}{5} + 34 = 66.4 + 34 = 100.4 \]<BR>\n" );
document.write( "So the demand equation is:<BR>\n" );
document.write( "\[ p = -\frac{2}{5}x + 100.4 \]\r<BR>\n" );
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document.write( "(c) Similarly, for the supply equation using the points \( (131, 34) \) and \( (181, 42) \), the slope is:<BR>\n" );
document.write( "\[ m = \frac{{42 - 34}}{{181 - 131}} = \frac{{8}}{{50}} = \frac{2}{5} \]<BR>\n" );
document.write( "And the y-intercept can be found as:<BR>\n" );
document.write( "\[ 34 = \frac{2}{5} \cdot 131 + b \]<BR>\n" );
document.write( "\[ b = \frac{2 \cdot 131}{5} + 34 = \frac{262}{5} + 34 = 52.4 + 34 = 86.4 \]<BR>\n" );
document.write( "So the supply equation is:<BR>\n" );
document.write( "\[ p = \frac{2}{5}x + 86.4 \]\r<BR>\n" );
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document.write( "(d) The equilibrium quantity and price are found where the demand and supply equations intersect:<BR>\n" );
document.write( "\[ -\frac{2}{5}x + 100.4 = \frac{2}{5}x + 86.4 \]<BR>\n" );
document.write( "Combining like terms:<BR>\n" );
document.write( "\[ -\frac{4}{5}x = -14 \]<BR>\n" );
document.write( "\[ x = \frac{14 \cdot 5}{4} = 17.5 \]<BR>\n" );
document.write( "Substituting back to find the price:<BR>\n" );
document.write( "\[ p = -\frac{2}{5} \cdot 17.5 + 100.4 = -7 + 100.4 = 93.4 \]\r<BR>\n" );
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document.write( "So the equilibrium occurs when the price of the clock is $93.4, and the quantity is 17.5.\r<BR>\n" );
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document.write( "Note: It seems like there may be an inconsistency in the given data, as the demand equation suggests a positive price for negative quantities and the supply equation suggests a positive price for quantities greater than 181. Make sure to double-check the provided data and the context in which these equations are being used. </SPAN>\n" );
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