document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=197063>Question 197063</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780495553458/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> How are addition and multiplication used to sovle a linear equation? Demonstrate by solving 15x + 7 = 31 + 3x<BR>\n" );
document.write( "I can sovle it 15x + 7 - 7 = 31 - 7 +3x<BR>\n" );
document.write( "               15x = 24 +3x<BR>\n" );
document.write( "               15x - 3x = 31 + 3x - 3x<BR>\n" );
document.write( "               12x = 24<BR>\n" );
document.write( "                 x = 2<BR>\n" );
document.write( "I dont know how to explain how are addition and multiplication used to solve a linear equation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #147731 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=147731\">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( "The idea is that you never subtract, rather you add the additive inverse, and you never divide, you multiply by the reciprocal (also called the multiplicative inverse).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 15x + 7 = 31 + 3x\">\r<BR>\n" );
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document.write( "<b>Add</b> -7 to both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 15x + 7 + (-7) = 31 + (-7) + 3x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 15x = 24 + 3x\">\r<BR>\n" );
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document.write( "<b>Add</b> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -3x\"> to both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 15x + (-3x) = 24 + 3x + (-3x)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 12x = 24\">\r<BR>\n" );
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document.write( "<b>Multiply</b> both sides by the reciprocal of the coefficient on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{12}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(\frac{1}{12}\right)\cdot 12x = \left(\frac{1}{12}\right)\cdot 24\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ x = 2\">\r<BR>\n" );
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document.write( "The point being that Addition and Multiplication are the two basic operations, while subtraction and division are simply inverse operations defined for convenience.  Both addition and multiplication are commutative over the real numbers, that is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ a + b = b + a\ \forall a,\,b\ \in\ \R\"> and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ ab = ba\ \forall a,\,b\ \in\ \R\">\r<BR>\n" );
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document.write( "But subtraction and division are <i>not</i> commutative, that is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \exists a,\,b\ \in\ \R\ |\ a - b \neq b - a\"> and \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \exists a,\,b\ \in\ \R\ |\ a \div b \neq b \div a\">\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" );
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