document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=112136>Question 112136</A>: <I><SPAN STYLE=\"word-break: break-all;\"> One number exceeds another by 5. One-fifth of their sum is five less than the smaller number. Find the number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #81818 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=81818\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The first sentence says that \"one number exceeds another ...\" This statement tells you that<BR>\n" );
document.write( "there are two unknown numbers. Call one of them x and the other y.  Since one number <BR>\n" );
document.write( "exceeds the other by 5, you know that if you take 5 away from one of the numbers, the result<BR>\n" );
document.write( "equals the other number. In equation form this can be written as:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+5+=+y ALIGN=MIDDLE  ALT=\"x+-+5+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== call this your first equation<BR>\n" );
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document.write( "In the next sentence the term \"of their sum\" tells you that you need to add the two numbers.<BR>\n" );
document.write( "Their sum is written as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+y ALIGN=MIDDLE  ALT=\"x+%2B+y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Then you need one-fifth of that sum, so you need to<BR>\n" );
document.write( "multiply the sum by one-fifth. In algebraic form this is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F5%29%28x+%2B+y%29 ALIGN=MIDDLE  ALT=\"%281%2F5%29%28x+%2B+y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== refer to this as the \"algebraic form\"<BR>\n" );
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document.write( "Finally you are told that this is algebraic form is 5 less than the smaller number.<BR>\n" );
document.write( "This means that if you add 5 to this algebraic form the result will equal the smaller<BR>\n" );
document.write( "number. From your first equation you can see that y is the smaller of the two numbers<BR>\n" );
document.write( "because you have to take 5 away from x to get y.  So add 5 to the algebraic<BR>\n" );
document.write( "form and the result will equal y. In equation form this is written as:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F5%29%28x+%2B+y%29+%2B+5+=+y ALIGN=MIDDLE  ALT=\"%281%2F5%29%28x+%2B+y%29+%2B+5+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > <=== call this your second equation<BR>\n" );
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document.write( "You now have two independent equations, each with two unknowns. Since there is a match between<BR>\n" );
document.write( "the number of equations and the number of unknowns (and the equations are independent<BR>\n" );
document.write( "of each other) you can solve for each unknown.  Your first equation gives you a value for y<BR>\n" );
document.write( "in terms of the second unknown ... which is x. You can see that y is equal to x minus 5.<BR>\n" );
document.write( "Therefore, you can go to your second equation and in it you can replace y with x - 5. (This<BR>\n" );
document.write( "method is referred to as solving the two equations by \"substitution.\") When you do this<BR>\n" );
document.write( "substitution, the second equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F5%29%28x+%2B+%28x+-+5%29%29+%2B+5+=+x+-+5 ALIGN=MIDDLE  ALT=\"%281%2F5%29%28x+%2B+%28x+-+5%29%29+%2B+5+=+x+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "On the left side the quantity (x + (x - 5)) can be simplified. Since the set of parentheses<BR>\n" );
document.write( "that contains x - 5 is preceded by a plus sign these parentheses can be removed without<BR>\n" );
document.write( "changing the signs of the x and the -5 inside. This leaves you with (x + x - 5) and when<BR>\n" );
document.write( "you add the x + x you end up with (2x - 5). Don't forget that this quantity has to be<BR>\n" );
document.write( "multiplied by one-fifth, so you now have reduced the left side and the equation is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F5%29%282x+-+5%29%29+%2B+5+=+x+-+5 ALIGN=MIDDLE  ALT=\"%281%2F5%29%282x+-+5%29%29+%2B+5+=+x+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can now get rid of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F5 ALIGN=MIDDLE  ALT=\"1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > by multiplying both sides of this equation (all terms)<BR>\n" );
document.write( "by 5. When you do this multiplication you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+-+5+%2B+25+=+5x+-+25 ALIGN=MIDDLE  ALT=\"2x+-+5+%2B+25+=+5x+-+25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This multiplication makes it a little easier because you don't have to mess with  fractions.<BR>\n" );
document.write( "On the left side combine the two numbers -5 and + 25 to get +20 and the equation then is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+20+=+5x+-+25 ALIGN=MIDDLE  ALT=\"2x+%2B+20+=+5x+-+25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Let's get rid of the 2x on the left side by subtracting 2x from both sides. When you do<BR>\n" );
document.write( "you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+20+=+3x+-+25 ALIGN=MIDDLE  ALT=\"+20+=+3x+-+25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then get rid of the -25 on the right side by adding 25 to both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45+=+3x ALIGN=MIDDLE  ALT=\"45+=+3x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can solve for x by dividing both sides by 3 and the result is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15+=+x ALIGN=MIDDLE  ALT=\"15+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You now know that x equals 15 and your very first equation told you that if you take 5<BR>\n" );
document.write( "away from x the result is y. Therefore, y equals 15 - 5 = 10.<BR>\n" );
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document.write( "So the answer to this problem is x = 15 and y = 10.<BR>\n" );
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document.write( "You can check the answer by returning to the two original equations, substituting <BR>\n" );
document.write( "15 for x and 10 for y, and then simplifying to ensure that the left side of the equation <BR>\n" );
document.write( "equals the right side.<BR>\n" );
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document.write( "Return to the first equation and when you substitute 15 for x and 10 for y you have:<BR>\n" );
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document.write( "15 - 5 = 10<BR>\n" );
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document.write( "That reduces to:<BR>\n" );
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document.write( "10 = 10<BR>\n" );
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document.write( "so that works.<BR>\n" );
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document.write( "Then go to the second equation and again substitute 15 for x and 10 for y to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F5%29%2815+%2B+10%29%2B5+=+10 ALIGN=MIDDLE  ALT=\"%281%2F5%29%2815+%2B+10%29%2B5+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Inside the parentheses on the left side the 15 + 10 becomes 25 ... so the equation<BR>\n" );
document.write( "reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F5%29%2825%29%2B5+=+10 ALIGN=MIDDLE  ALT=\"%281%2F5%29%2825%29%2B5+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and one-fifth of 25 is 5 so the equation reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+%2B+5+=+10 ALIGN=MIDDLE  ALT=\"5+%2B+5+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "which simplifies to<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10+=+10 ALIGN=MIDDLE  ALT=\"10+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Both your first and second equations work out if x = 15 and y = 10. Therefore, your answers<BR>\n" );
document.write( "are correct.<BR>\n" );
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document.write( "Hope this helps you to understand the problem and see how it can be worked out to get<BR>\n" );
document.write( "the answer.<BR>\n" );
document.write( ".\r<BR>\n" );
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document.write( " </SPAN>\n" );
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