document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=334642>Question 334642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what does it mean for a number to be the \"solution\" to an equation?<BR>\n" );
document.write( "please give me an example of a variable equation and its solution and please explain in complete sentences how you know that number is the solution. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #239806 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=239806\">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( "A solution of an equation is any entity that can be substituted for the variable or variables such that the the equation is a true statement.\r<BR>\n" );
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document.write( "For example, consider the equation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ +\ 3\ =\ 8\">\r<BR>\n" );
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document.write( "It should be clear that if you replace the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 5\">, then you have a true statement because 5 plus 3 does indeed equal 8.  But if you replace <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> with any other number, then you have a false statement.  Let's see what happens if we use <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2\">.  2 plus 3 is 5 which is most assuredly not equal to 8.  And the same thing would happen for any other number besides 5 substituted for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.\r<BR>\n" );
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document.write( "Point of semantics:  You said \"...number to be <b>the</b> \"solution\"...\"\r<BR>\n" );
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document.write( "Using the word <b>the</b> in that context is only appropriate when you have single variable linear equations.  It is more appropriate to say \"...a solution\" because, in general, there may be more than one.  For example, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\ +\ 2x\ -\ 3\ =\ 0\"> has two solutions, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -3\"> (try it and see for yourself).\r<BR>\n" );
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document.write( "Furthermore, some equations have solutions that are not numbers, but ordered pairs of numbers.  You see this when you have two-variable equations.  For instance, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2x\ -\ y\ =\ 3\">, has an infinite number of solutions one of which is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (2,1)\">.\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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