document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=100248>Question 100248</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of two numbers equal 14 and the difference is 6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #72999 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=72999\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You have two unknown numbers. Represent them by X and Y.<BR>\n" );
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document.write( "The problem tells you that the sum of these two numbers is 14. Represent this in equation<BR>\n" );
document.write( "form as:<BR>\n" );
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document.write( "X + Y = 14<BR>\n" );
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document.write( "The problem also tells you that the difference of these two numbers is 6. Represent this<BR>\n" );
document.write( "in equation form as:<BR>\n" );
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document.write( "X - Y = 6<BR>\n" );
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document.write( "There are several ways to solve this pair of equations. One way we can do it is by variable<BR>\n" );
document.write( "elimination. Write the two equations together as follows:<BR>\n" );
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document.write( "X + Y = 14<BR>\n" );
document.write( "X - Y = 6<BR>\n" );
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document.write( "Notice that if we add the two equations vertically in columns we get X + X = 2X and for<BR>\n" );
document.write( "the next column the +Y adds to -Y to cancel each other out. Finally notice on the other<BR>\n" );
document.write( "side of the equal sign that the 14 adds to the 6 to give 20. So when we add the two equations<BR>\n" );
document.write( "vertically we are left with the resulting equation:<BR>\n" );
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document.write( "2X = 20<BR>\n" );
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document.write( "We can now solve for X by dividing both sides by 2:<BR>\n" );
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document.write( "2X/2 = 20/2 <BR>\n" );
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document.write( "And after the division we are left with:<BR>\n" );
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document.write( "X = 10<BR>\n" );
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document.write( "So we have one of the unknown numbers identified as 10. Since we know that the sum of the<BR>\n" );
document.write( "two numbers must be 14, the missing 4 has to be the other number. This makes the two numbers<BR>\n" );
document.write( "10 and 4. Notice that the difference in the two numbers is 10 - 4 and that equals 6 <BR>\n" );
document.write( "just as the problem said it should. This is just another check to ensure that our answer<BR>\n" );
document.write( "is correct. The answers to the problem you posted are 10 and 4. Now for a lesson in using the<BR>\n" );
document.write( "process of variable elimination to solve a pair of equations that have two unknowns.<BR>\n" );
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document.write( "The reason that this method worked is that we can't solve a single equation for two unknowns.<BR>\n" );
document.write( "If there are two independent equations for two unknowns, we should be able to solve them<BR>\n" );
document.write( "if there is a single but common solution. Because of the values of the unknowns in the two<BR>\n" );
document.write( "equations we originally wrote, we noticed that we could add the equations and one of the<BR>\n" );
document.write( "variables (the Y) would cancel out so that we would be left with just one equation that<BR>\n" );
document.write( "had one unknown. If the two equations did not have one of the variables equal but of <BR>\n" );
document.write( "opposite sign to the same variable in the other equation, we would have to make it that way <BR>\n" );
document.write( "in order to use variable elimination.<BR>\n" );
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document.write( "For example only to show you how variable elimination works... suppose the two equations had been:<BR>\n" );
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document.write( "2X + 6Y = 20 and<BR>\n" );
document.write( "3X + 4Y = 10<BR>\n" );
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document.write( "We could multiply the entire top equation (all terms) by 3 to convert it to:<BR>\n" );
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document.write( "6X + 18Y = 60<BR>\n" );
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document.write( "and we could multiply both sides of the bottom equation (all terms) by -2 to make it<BR>\n" );
document.write( "become:<BR>\n" );
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document.write( "-6X - 8Y = -20<BR>\n" );
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document.write( "So we have converted the original pair of equations to:<BR>\n" );
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document.write( "+6X + 18Y = 60 and<BR>\n" );
document.write( "-6X - 8Y = -20<BR>\n" );
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document.write( "We have made the terms containing X equal but opposite in sign. Notice that if we now<BR>\n" );
document.write( "add the two equations vertically in columns that the +6X and the -6X will cancel each<BR>\n" );
document.write( "other and the rest of the sums will leave us +18Y - 8Y = 10Y and on the other side we<BR>\n" );
document.write( "have +60 -20 = +40. So we have reduced the pair of equations to:<BR>\n" );
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document.write( "10Y = 40<BR>\n" );
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document.write( "And dividing both sides of that equation by 10 gives us Y = 4<BR>\n" );
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document.write( "Now that we know Y is equal to 4, we can return to either of the original equations<BR>\n" );
document.write( "in this example and substitute 4 for Y and solve for X. One of the original two equations<BR>\n" );
document.write( "we started with was:<BR>\n" );
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document.write( "3X + 4Y = 10<BR>\n" );
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document.write( "Substituting 4 for Y results in:<BR>\n" );
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document.write( "3X + 4(4) = 10<BR>\n" );
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document.write( "And multiplying the 4 times 4 converts this to:<BR>\n" );
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document.write( "3X + 16 = 10<BR>\n" );
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document.write( "Getting rid of the +16 on the left side by subtracting 16 from both sides we are left with:<BR>\n" );
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document.write( "3X = -6<BR>\n" );
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document.write( "and we can solve for X by dividing both sides by 3 to get X = -2. So the answers to this sample<BR>\n" );
document.write( "problem are X = -2 and Y = +4.<BR>\n" );
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document.write( "Hope this helps you to understand the problem you posted and gives you some insight into<BR>\n" );
document.write( "solving two equations with two unknowns by using the process of eliminating a variable<BR>\n" );
document.write( "from the pair of equations.<BR>\n" );
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