document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=335237>Question 335237</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please assist me in solving this system by the substitution method as well as determining whether the equations are independent, dependent, or inconsistent.\r<BR>\n" );
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document.write( "x + 3y = 2<BR>\n" );
document.write( "-x +y = 1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #240281 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=240281\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left\{\ x\ +\ 3y\ =\ 2\cr-x\ +\ \ y\ =\ 1\right\">\r<BR>\n" );
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document.write( "Solve either of the equations for either of the variables in terms of the other.  I choose to solve equation 1 for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ +\ 3y\ =\ 2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ -3y\ +\ 2\">\r<BR>\n" );
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document.write( "Use the RHS expression that is equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> as a substitution into the 2nd equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -\left(-3y\ +\ 2\right)\ +\ \ y\ =\ 1\">\r<BR>\n" );
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document.write( "Solve this equation for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">.  If this substitution produces a value for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large  y\"> you can substitute back into either of the original equations to solve for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.  If this substitution produces a trivial identity, such as 0 = 0, then you have infinite solutions.  If this substitution produces an absurdity, such as 3 = 0, then you have no solutions. \r<BR>\n" );
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document.write( "Consistent systems have at least one solution.\r<BR>\n" );
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document.write( "Inconsistent systems have no solutions.\r<BR>\n" );
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document.write( "Independent systems have exactly one solution.\r<BR>\n" );
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document.write( "Dependent systems have infinite solutions.\r<BR>\n" );
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document.write( "A system that represents a pair of lines that intersect in one point is consistent and independent.  A system that represents a pair of lines that graph to the same line is consistent and dependent.  A system that represents a pair of parallel lines is inconsistent.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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