1.1

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=2 height=24 BGCOLOR="#003366" TEXT="#F7F7FF"> <div> <FONT SIZE="4" COLOR="#00FFCC" FACE="Bell MT" ><B>P r e c á l c u l o 21</B></FONT><FONT SIZE="4" COLOR="#FFFFFF" FACE="Bell MT" ><B> |</B></FONT><FONT SIZE="3" COLOR="#00FFCC" FACE="Bell MT" ><B> </B></FONT><A HREF="index.htm" TARGET="_top" TITLE="Precálculo21"><U><B>Página principal</B></U></A><B>  |  </B><A HREF="id418.htm" TARGET="_top" TITLE="_"><U><B>Contenido</B></U></A><B>  | </B><A HREF="id492.htm" TARGET="_top" TITLE="Álgebra y trigonometría de Sullivan"><U><B>Álgebra y trigonometría de Sullivan</B></U></A></div> <DIV ALIGN="LEFT"></DIV> <div> </div> </td> </tr><tr> <td valign="top" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3073775660"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> <bR> </td> <td valign="top" height=456 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Ejercicios 1.1</B></div> <div> <B>Ecuaciones lineales</B></div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:728px;height:90px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="9294650114"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <div> <B>Procedimiento </B></div> <div> Al resolver una ecuación lineal, "se despeja la incógnita", esto es se aisla la variable que representa a la incógnita en el miembro izquierdo de la igualdad y en el miembro derecho se da su valor numérico. En el proceso se escriben una serie de ecuaciones equivalentes hasta despejar la incógnita y simplificar el miembro derecho:</div> <div> <B>1.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se hacen las restricciones necesarias a la incógnita (se descartan los valores que producen división por 0).</FONT></div> <div> <B>2.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Si la incógnita sólo aparece en el miembro derecho, se hace una transposición de miembros.</FONT></div> <div> <B>3.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Si es el caso, se efectúan las operaciones indicadas (aplicando la propiedad distributiva, etc). Si hay términos fraccionarios, se eliminan los denominadores multiplicando ambos miembros de la ecuación por el mínimo común múltiplo de los denominadores (MCD). Si hay coeficientes que son fracciones decimales, se multiplican todos los términos por una potencia de 10 igual al mayor número de cifras decimales que se presente en algún coeficiente.</FONT></div> <div> <B>4.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se reducen términos semejantes en cada miembro de la ecuación.</FONT></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B>5.  </B></FONT>Se efectúa una transposición de términos: los que tienen la incógnita <I>x</I> se escriben en el miembro izquierdo de la igualdad y los otros términos en el miembro derecho. Aquellos términos que pasan de un lado de la igualdad al otro cambian de signo los que permanecen en su lado mantienen su signo.</div> <div> <B>6.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se reducen términos semejantes.</FONT></div> <div> <FONT SIZE="3" COLOR="#0099FF" FACE="Bell MT" ><B>7.  </B></FONT>Se culmina de despejar la incógnita, dividiendo ambos miembros entre el coeficiente de la incógnita <I>x</I>.</div> <div> <B>8.  </B><FONT SIZE="3" COLOR="#000000" FACE="Bell MT" >Se verifica la solución obtenida: se sustituye, en la ecuación original, el valor numérico obtenido para la incógnita; se efectúan las operaciones indicadas, se simplifica y se observa si se halla una igualdad cierta.</FONT></div> <div> <B>Enunciados de los ejercicios y problemas y procedimiento que se sigue (pulse sobre el ícono de la imagen <IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> o el video </B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="12da18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><B>correspondiente para que observe la solución que se da paso a paso):</B></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >17</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >64</FONT>, resuelva cada ecuación:</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Bell MT" ><IMG SRC="131363130.gif" border=0 width="99" height="19" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#17" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13157c130.gif" border=0 width="124" height="19" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#21" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="1318a71b0.gif" border=0 width="167" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#25" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="131cb41b0.gif" border=0 width="180" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#27" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="131f90290.gif" border=0 width="144" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#29" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <B><IMG SRC="132275290.gif" border=0 width="117" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#31" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>        <IMG SRC="13257c290.gif" border=0 width="124" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#33" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13298f130.gif" border=0 width="143" height="19" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#35" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="132c8f290.gif" border=0 width="143" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#37" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="132f6d2d0.gif" border=0 width="109" height="45" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#40" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="133264290.gif" border=0 width="100" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#42" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="1335c01d0.gif" border=0 width="192" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#44" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="13389d1b0.gif" border=0 width="157" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#48" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="133bbf290.gif" border=0 width="191" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B><A HREF="#52" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="133e80290.gif" border=0 width="128" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#55" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="134197290.gif" border=0 width="151" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#58" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="1344bd290.gif" border=0 width="189" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#62" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="134be9290.gif" border=0 width="233" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#64" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>E</B></FONT>n los problemas <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >69</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >74</FONT>, resuelva cada ecuación. Las letras <I>a</I>, <I>b</I> y <I>c</I> son constantes:</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Bell MT" ><IMG SRC="134e8f150.gif" border=0 width="143" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#69" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="1351ef290.gif" border=0 width="239" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#71" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="1354a9290.gif" border=0 width="169" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#73" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="1357b1150.gif" border=0 width="689" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#76" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Bell MT" ><B>L</B></FONT>os problemas <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >77</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Bell MT" >82 </FONT>dan algunas fórmulas que ocurren en las aplicaciones. Resuelva cada fórmula para la variable indicada:</div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Bell MT" ><IMG SRC="135a132d0.gif" border=0 width="275" height="45" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#77" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="135dfb1b0.gif" border=0 width="251" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#78" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="1360e92b0.gif" border=0 width="233" height="43" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#79" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="1363ec150.gif" border=0 width="236" height="21" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#80" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="136601290.gif" border=0 width="257" height="41" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#81" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A><B>       <IMG SRC="1369fb180.gif" border=0 width="251" height="24" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#82" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <B><IMG SRC="136c3f2b0.gif" border=0 width="831" height="43" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#83" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <B><IMG SRC="136f51380.gif" border=0 width="849" height="56" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#85" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <B><IMG SRC="13725f3d0.gif" border=0 width="863" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#86" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <div> <B><IMG SRC="13753d2d0.gif" border=0 width="829" height="45" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </B><A HREF="#87" TITLE="1.1"><U><B><IMG SRC="12db18180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></U></A></div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <B>  Soluciones</B></div> <div>  <A NAME="17"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>17</div> <div> <IMG SRC="131254b30.gif" border=0 width="596" height="179" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="21"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>21</div> <div> <IMG SRC="131608c70.gif" border=0 width="776" height="199" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="25"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>25</div> <div> <IMG SRC="131929bc0.gif" border=0 width="809" height="188" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="27"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>27</div> <div> <IMG SRC="131b54130.gif" border=0 width="852" height="275" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="29"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>29</div> <div> <IMG SRC="131ef0390.gif" border=0 width="752" height="313" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="31"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>31</div> <div> <IMG SRC="1321f8240.gif" border=0 width="760" height="292" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="33"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>33</div> <div> <IMG SRC="1324070b0.gif" border=0 width="775" height="267" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="35"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>35</div> <div> <IMG SRC="1328bff50.gif" border=0 width="703" height="245" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="37"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>37</div> <div> <IMG SRC="132b25830.gif" border=0 width="805" height="387" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="40"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>40</div> <div> <IMG SRC="132e254b0.gif" border=0 width="805" height="331" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="42"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>42</div> <div> <IMG SRC="13311f350.gif" border=0 width="799" height="309" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="44"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>44</div> <div> <IMG SRC="133473730.gif" border=0 width="883" height="371" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="48"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>48</div> <div> <IMG SRC="133738cd0.gif" border=0 width="824" height="205" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="52"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>52</div> <div> <IMG SRC="133a70990.gif" border=0 width="880" height="409" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="55"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>55</div> <div> <IMG SRC="133d48580.gif" border=0 width="840" height="344" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="58"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>58</div> <div> <IMG SRC="134074fc0.gif" border=0 width="884" height="508" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="62"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>62</div> <div> <IMG SRC="134374ed0.gif" border=0 width="884" height="493" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="64"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>64</div> <div> <IMG SRC="134a70c70.gif" border=0 width="880" height="455" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="69"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>6<FONT SIZE="4" COLOR="#FFFFFF" FACE="Bell MT" >9</FONT></div> <div> <IMG SRC="134dada00.gif" border=0 width="685" height="160" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="71"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>71</div> <div> <IMG SRC="135000e30.gif" border=0 width="768" height="227" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>73</div> <div> <IMG SRC="1353603b0.gif" border=0 width="864" height="315" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="76"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>76</div> <div> <IMG SRC="1356b1cd0.gif" border=0 width="689" height="205" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="77"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>77</div> <div> <IMG SRC="135939100.gif" border=0 width="825" height="272" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="78"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>78</div> <div> <IMG SRC="135c695d0.gif" border=0 width="617" height="93" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="79"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>79</div> <div> <IMG SRC="135f396d0.gif" border=0 width="313" height="109" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="80"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>80</div> <div> <IMG SRC="136247590.gif" border=0 width="327" height="89" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="81"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>81</div> <div> <IMG SRC="1365456d0.gif" border=0 width="581" height="109" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="82"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>82</div> <div> <IMG SRC="1368845d0.gif" border=0 width="388" height="93" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="83"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>83</div> <div> <IMG SRC="136b3f2d0.gif" border=0 width="831" height="301" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="85"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>85</div> <div> <IMG SRC="136e51c00.gif" border=0 width="849" height="448" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="86"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>86</div> <div> <IMG SRC="13715fcc0.gif" border=0 width="863" height="460" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="87"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>87</div> <div> <IMG SRC="13743d4c0.gif" border=0 width="829" height="332" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> </td> </tr></table></body>