document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=70768>Question 70768</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0131492594/algebrahomeworkh>Introductory and intermediate algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> The sum of two numbers is 2.  If one number is subtracted from the other, their difference is 8.  Find the numbers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #50522 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=50522\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The first sentence and the last sentence in this problem tell you that you have two unknown<BR>\n" );
document.write( "numbers to find. So let's call them by different letters.  Normally we would use x and y,<BR>\n" );
document.write( "but really we could use any two letters. Just to show you how to work with other letters,<BR>\n" );
document.write( "let's call these two numbers A and B.<BR>\n" );
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document.write( "The problem tells you that when you add the two numbers, the answer is 2.  So let's write<BR>\n" );
document.write( "that as an equation. Adding would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+%2B+B ALIGN=MIDDLE  ALT=\"A+%2B+B\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  In equation form<BR>\n" );
document.write( "that is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+%2B+B+=+2 ALIGN=MIDDLE  ALT=\"A+%2B+B+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next we are told that if one of the numbers is subtracted from the other, the result is 8.<BR>\n" );
document.write( "So let's pick one of the numbers and subtract it from the other. Let's subtract B from A.<BR>\n" );
document.write( "In algebraic form this would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+-+B ALIGN=MIDDLE  ALT=\"A+-+B\"   DISABLED_style= \"max-width:90%; height:auto;\" > and since this is to equal <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can<BR>\n" );
document.write( "write another equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+-+B+=+8 ALIGN=MIDDLE  ALT=\"A+-+B+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "We now have two equations with two unknowns.  The fact that there are two independent<BR>\n" );
document.write( "equations means that we can solve for the two unknowns.  There are several ways to solve <BR>\n" );
document.write( "them.  Let's use a way called substitution.  We'll solve one of the equations for one<BR>\n" );
document.write( "of the unknowns in terms of the other, and then we'll substitute that answer into the<BR>\n" );
document.write( "other equation to solve it.  Hard to explain, but fairly easy to do. Let's try solving<BR>\n" );
document.write( "the second equation for A.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A-B+=+8 ALIGN=MIDDLE  ALT=\"A-B+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "What happens if we add B to both sides of this equation? [Remember whatever we do to one<BR>\n" );
document.write( "side of an equation ... adding, subtracting, multiplying, dividing ... we also have to <BR>\n" );
document.write( "do to the other side.] When we add +B to the left side it cancels out the -B and the left<BR>\n" );
document.write( "side becomes just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  And when we add +B to the right side, the right side becomes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B+%2B+8 ALIGN=MIDDLE  ALT=\"B+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  So we have solved the second equation for A. It is just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+B%2B8 ALIGN=MIDDLE  ALT=\"A+=+B%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now let's go back to our first equation which was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BB=2 ALIGN=MIDDLE  ALT=\"A%2BB=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. From our second equation<BR>\n" );
document.write( "we found that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > was the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2B8 ALIGN=MIDDLE  ALT=\"B%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So in our first equation we can replace the<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2B8 ALIGN=MIDDLE  ALT=\"B%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. When we do that the first equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2B8%2BB+=+2 ALIGN=MIDDLE  ALT=\"B%2B8%2BB+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Adding the B terms together on the left side gives:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2B+%2B+8+=2 ALIGN=MIDDLE  ALT=\"2B+%2B+8+=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next add a negative 8 on the left side to get rid of the 8 on the left side. But if we do<BR>\n" );
document.write( "that we must also add a negative 8 on the right side. The right side becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8%2B2 ALIGN=MIDDLE  ALT=\"-8%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So now our equation is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2B+=+-6 ALIGN=MIDDLE  ALT=\"2B+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "To solve for just B we divide both sides by 2.  The left side becomes just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the <BR>\n" );
document.write( "right side is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%2F2 ALIGN=MIDDLE  ALT=\"-6%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So finally we can write this equation<BR>\n" );
document.write( "as:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B+=+-3 ALIGN=MIDDLE  ALT=\"B+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "We have one of the unknown numbers! Now we can return to either of the original equations.<BR>\n" );
document.write( "For example, let's return to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BB=2 ALIGN=MIDDLE  ALT=\"A%2BB=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. But we now know that B is -3. So let's substitute<BR>\n" );
document.write( "-3 for B in this equation.  When we do, the equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+%2B+%28-3%29=+2 ALIGN=MIDDLE  ALT=\"A+%2B+%28-3%29=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "which becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+-+3+=+2 ALIGN=MIDDLE  ALT=\"A+-+3+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "To get rid of the -3 on the left side we add +3 to both sides.  The result is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+5 ALIGN=MIDDLE  ALT=\"A+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "That's the last thing we had to do. We can now say that the two numbers are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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( ".<BR>\n" );
document.write( "Hope this helps you see a way of working problems such as these. As you see, you will need<BR>\n" );
document.write( "to be familiar with the rules for algebraic addition, subtraction, multiplication,<BR>\n" );
document.write( "and division to work a problem like this.  So make sure you learn those rules well.<BR>\n" );
document.write( " \r<BR>\n" );
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document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );