document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=800872>Question 800872</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is a quick and easy way of solving algebraic equations? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #483369 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=483369\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It depends on the type of equation.<BR>\n" );
document.write( "That's the quick and short answer.<BR>\n" );
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document.write( "I'll give you a long answer.<BR>\n" );
document.write( "Algebra is a language.<BR>\n" );
document.write( "To use algebra you need to know the meaning of symbols and you need to know the rules for order of operations. Those rules are needed so we can understand one another.<BR>\n" );
document.write( "To be understood in any language, you need to form sentences arranging words in a certain order. If you scramble the order of the words in a sentence, your meaning is likely to be misunderstood.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The simplest equations are equations like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2B3x=23 ALIGN=MIDDLE  ALT=\"5%2B3x=23\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5=23 ALIGN=MIDDLE  ALT=\"3x%2B5=23\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "According to the rules for order of operations,<BR>\n" );
document.write( "the expressions <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2B3x ALIGN=MIDDLE  ALT=\"5%2B3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5 ALIGN=MIDDLE  ALT=\"3x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "both mean that the value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is first multiplied times 3,<BR>\n" );
document.write( "and then 5 is added.<BR>\n" );
document.write( "The equations say that the result of doing that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23 ALIGN=MIDDLE  ALT=\"23\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "To solve that equation (to get back to the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >),<BR>\n" );
document.write( "you need to undo the steps in reverse order.<BR>\n" );
document.write( "That's how things are usually undone in life.<BR>\n" );
document.write( "You take off your shoes before you take of your socks,<BR>\n" );
document.write( "undoing the putting on of socks and shoes in the reverse order.<BR>\n" );
document.write( " <BR>\n" );
document.write( "So you transform the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5=23 ALIGN=MIDDLE  ALT=\"3x%2B5=23\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=23-5 ALIGN=MIDDLE  ALT=\"3x=23-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > by undoing the adding of 5.<BR>\n" );
document.write( "That simplifies into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=18 ALIGN=MIDDLE  ALT=\"3x=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The next step is undoing the multiplication times 3,<BR>\n" );
document.write( "transforming <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=18 ALIGN=MIDDLE  ALT=\"3x=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=18 ALIGN=MIDDLE  ALT=\"x=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >÷<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "which simplifies into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "At each step you can think of the equation as the two expressions on both sides of the equal sign being balanced on the pans of a two pan balance,<BR>\n" );
document.write( "or on the two ends of a seesaw.<BR>\n" );
document.write( "For the two sides to stay equal (balanced),<BR>\n" );
document.write( "whatever you do to one side (the entire side),<BR>\n" );
document.write( "you have to do to the other side.<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5=23 ALIGN=MIDDLE  ALT=\"3x%2B5=23\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=18 ALIGN=MIDDLE  ALT=\"3x=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > by subtracting 5 from both sides.<BR>\n" );
document.write( "I can write that in steps as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5=23 ALIGN=MIDDLE  ALT=\"3x%2B5=23\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5-5=23-5 ALIGN=MIDDLE  ALT=\"3x%2B5-5=23-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=23-5 ALIGN=MIDDLE  ALT=\"3x=23-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=18 ALIGN=MIDDLE  ALT=\"3x=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "or I can just write<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5=23 ALIGN=MIDDLE  ALT=\"3x%2B5=23\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=18 ALIGN=MIDDLE  ALT=\"3x=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You need to write as many steps as needed for your comfort and the teachers comfort.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "It is a good idea to verify the solution you find.<BR>\n" );
document.write( "To verify, you substitute your solution and see if it makes the original equation true.<BR>\n" );
document.write( "To verify that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5=23 ALIGN=MIDDLE  ALT=\"3x%2B5=23\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "I substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5 ALIGN=MIDDLE  ALT=\"3x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "It turns into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%286%29%2B5 ALIGN=MIDDLE  ALT=\"3%286%29%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2B5%2B23 ALIGN=MIDDLE  ALT=\"18%2B5%2B23\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > makes the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5=23 ALIGN=MIDDLE  ALT=\"3x%2B5=23\"   DISABLED_style= \"max-width:90%; height:auto;\" > true.<BR>\n" );
document.write( "That means that my solution is correct.<BR>\n" );
document.write( "I did not make a mistake this time. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );