Problemas 1.1

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</script>  </div> </td> <td valign="top" height=432 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>Problemas 1.1</B></div> <div> <B>Dos ecuaciones lineales con dos incógnitas</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-1578463498418701" data-ad-slot="5388515922"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <div>  <A NAME="apuntes"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>apuntes</div> <div> <TABLE BORDER="0" CELLPADDING="4" CELLSPACING="1" WIDTH="904" BORDERCOLORLIGHT="#006666" BORDERCOLORDARK="#006666" FRAME="VOID" RULES="NONE" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#006666 HEIGHT= 25 WIDTH="878"><div> <B>Apuntes</B></div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 45 WIDTH="585"><div> <A HREF="#sistema_2x2" TITLE="Problemas 1.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Sistema de ecuaciones lineales 2x2</U></A></div> </tD> <tD VALIGN=TOP WIDTH="292"><div> <A HREF="#teorema_1_1_1" TITLE="Problemas 1.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Teorema 1.1.1</U></A> (Teorema de resumen)</div> </tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 23 ><NOBR><div> <A HREF="#propiedades_del_algebra_elemental" TITLE="Problemas 1.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Dos propiedades fundamentales del álgebra elemental</U></A></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <A HREF="#sistemas_equivalentes" TITLE="Problemas 1.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Sistemas equivalentes</U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 26 WIDTH="585"><div> <A HREF="#sistema_incosnsistente" TITLE="Problemas 1.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Sistemas consistentes e inconsistentes</U></A></div> </tD> <tD VALIGN=TOP WIDTH="292"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="292" HSPACE="0" VSPACE="0"></tD> </tR> <tR> <tD VALIGN=TOP BGCOLOR=#CCFFFF HEIGHT= 26 ><NOBR><div> <A HREF="#enunciados_1" TITLE="Problemas 1.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Enunciados de los problemas </U></A></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="292"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="26" ALIGN="bottom" WIDTH="292" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP BGCOLOR=#CCFFCC><NOBR><div> <A HREF="#soluciones" TITLE="Problemas 1.1"><U><IMG BORDER="0" SRC="Symb075c00b7011866660000.gif" HEIGHT="19" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Soluciones de los problemas </U></A></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#006666 HEIGHT= 20 WIDTH="878"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="20" ALIGN="bottom" WIDTH="878" HSPACE="0" VSPACE="0"></tD> </tR> </TABLE></div> <div>  <A NAME="sistema_2x2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>sistema 2x2</div> <div> <B>Dos ecuaciones lineales con dos incógnitas.</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > La forma general de un sistema de dos ecuaciones con dos incógnitas es:</FONT></div> <DIV ALIGN="LEFT"></DIV> <div> <IMG SRC="5538c3d0.gif" border=0 width="140" height="61" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="5548d1d0.gif" border=0 width="653" height="29" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="propiedades_del_algebra_elemental"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Propiedades del algebra elemental</div> <div> <B>Propiedades del álgebra el</B><B>em</B><B>ental.</B> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > Dos propiedades importantes del álgebra elemental que se aplican al resolver un sistema de ecuaciones lineales son:</FONT></div> <div> <FONT SIZE="4" COLOR="#BF00BF" FACE="Bell MT" ><B>Propiedad A:</B></FONT><FONT SIZE="4" COLOR="#FF00FF" FACE="Bell MT" ><B>   </B></FONT>Si <I>a </I>= <I>b  </I>y  <I>c </I>= <I>d</I>, entonces <I>a </I>+ <I>c  </I>= <I>b +</I> <I>d.</I></div> <div> <FONT SIZE="4" COLOR="#BF00BF" FACE="Bell MT" ><B>Propiedad B:</B></FONT><FONT SIZE="4" COLOR="#FF00FF" FACE="Bell MT" ><B>   </B></FONT>Si <I>a </I>= <I>b </I>y <I>c </I>es cualquier número real, entonces <I>ca </I>= <I>cb</I>.</div> <div> La propiedad A establece que si se suman dos ecuaciones se obtiene una tercera ecuación correcta. La propiedad B establece que si se multiplican ambos lados de una ecuación por una <FONT SIZE="3" COLOR="#000000" FACE="TimesNewRomanMTStd" >constante</FONT> se obtiene una segunda ecuación válida.</div> <div>  <A NAME="teorema_1_1_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Teorema 1.1.1</div> <div> <IMG SRC="3dd9dfd0.gif" border=0 width="925" height="253" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="sistema_incosnsistente"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>Sistema incosnsistente</div> <div> <B>Sistemas consistentes e incosistentes.</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >  Un sistema de ecuaciones que no tiene solución se dice que es </FONT><I>inconsistente</I><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >. En cambio, si el sistema tiene solución única o un número infinito de soluciones se dice que es </FONT><I>consi</I><I>ste</I><I>nte</I><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >.</FONT></div> <div>  <A NAME="sistemas_equivalentes"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="1" COLOR="#FFFFFF" FACE="Bell MT" >Sistemas equivalentes</FONT></div> <div> <B>Sistemas equivalentes</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >.  Dos sistemas de ecuaciones consistentes son </FONT><I>equivalentes</I><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > cuando tienen el mismo conjunto solución.</FONT></div> <DIV ALIGN="LEFT"></DIV> <div>  <A NAME="enunciados_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><FONT SIZE="1" COLOR="#FFFFFF" FACE="Bell MT" >Enunciados</FONT></div> <div> <B>Enunciados de los Problemas 1.1</B></div> <bR> <div> <B>          S</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >olución en imagen <IMG SRC="12718180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> y o video <IMG SRC="12818180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> de los Problemas 1.1:</FONT></div> <div> <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B> <A NAME="1_a_12"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B></FONT><B>E</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >n los problemas del </FONT>1<FONT SIZE="4" COLOR="#000000" FACE="Bell MT" > al </FONT>18<FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >, encuentre todas las soluciones (si existen) a los sistemas dados. En cada caso calcule el valor de:</FONT></div> <div> <IMG SRC="19b84190.gif" border=0 width="132" height="25" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <FONT SIZE="3" COLOR="#000000" FACE="Bell MT" ><IMG SRC="12973340.gif" border=0 width="115" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> </FONT><A HREF="#1" TITLE="Problemas 1.1"><U><IMG SRC="12718180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/st_ATzjLk34" TARGET="_top" TITLE="https://youtu.be/st_ATzjLk34"><U><IMG SRC="12818180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="5b08c340.gif" border=0 width="140" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/st_ATzjLk34" TARGET="_top" TITLE="https://youtu.be/st_ATzjLk34"><U><IMG SRC="12818180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="5b178340.gif" border=0 width="120" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/st_ATzjLk34" TARGET="_top" TITLE="https://youtu.be/st_ATzjLk34"><U><IMG SRC="12818180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="12b6b340.gif" border=0 width="107" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#4" TITLE="Problemas 1.1"><U><IMG SRC="12718180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/Jj02AdV5zLY" TARGET="_top" TITLE="https://youtu.be/Jj02AdV5zLY"><U><IMG SRC="12818180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="12d7f340.gif" border=0 width="127" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#5" TITLE="Problemas 1.1"><U><IMG SRC="12718180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/Jj02AdV5zLY" TARGET="_top" TITLE="https://youtu.be/Jj02AdV5zLY"><U><IMG SRC="12818180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="5b47f340.gif" border=0 width="127" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><IMG SRC="12818180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT>       <IMG SRC="19c81340.gif" border=0 width="129" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <A HREF="#7" TITLE="Problemas 1.1"><U><IMG SRC="12718180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/RWRcpK4w-JE" TARGET="_top" TITLE="https://youtu.be/RWRcpK4w-JE"><U><IMG SRC="12818180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="5b893340.gif" border=0 width="147" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/RWRcpK4w-JE" TARGET="_top" TITLE="https://youtu.be/RWRcpK4w-JE"><U><IMG SRC="12818180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="5b9a7340.gif" border=0 width="167" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/RWRcpK4w-JE" TARGET="_top" TITLE="https://youtu.be/RWRcpK4w-JE"><U><IMG SRC="12818180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A>       </FONT><IMG SRC="5baa7340.gif" border=0 width="167" height="52" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/jZj9l7jtdQs" TARGET="_top" TITLE="https://youtu.be/jZj9l7jtdQs"><U><IMG SRC="12818180.jpg" bo