document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=771430>Question 771430</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A four-digit integer WXYZ, in which W, X, Y, and Z each represent a different digit, is formed according to the following rules\r<BR>\n" );
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document.write( "1.) X = W + Z + Y<BR>\n" );
document.write( "2.) W = Y +1<BR>\n" );
document.write( "3.) Z = W - 5\r<BR>\n" );
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document.write( "What is the four-digit integer?\r<BR>\n" );
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document.write( "I tried plugging in, say, Y + 1 for W in the first equation, then W - 5 for Z and Y + 1 for W, trying this method in the other equations, but I haven't gotten anywhere with this.\r<BR>\n" );
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document.write( "Could you please help me find a way to solve this?  If it helps, the answer is 5940.  I'm practicing for the SAT coming up.\r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #470227 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39617)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=470227\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A method that worked was use the w formula in the other two equations to eliminate it through substitution:\r<BR>\n" );
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document.write( "Leaving out some of the steps,<BR>\n" );
document.write( "x-2y-z=1 and z-y=-4\r<BR>\n" );
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document.write( "Next, let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=k ALIGN=MIDDLE  ALT=\"z=k\"   DISABLED_style= \"max-width:90%; height:auto;\" >, some as yet not known constant,<BR>\n" );
document.write( "and putting into a well chosen form,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y-k=1 ALIGN=MIDDLE  ALT=\"x-2y-k=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z-y=-4 ALIGN=MIDDLE  ALT=\"z-y=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "then using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2y=k%2B1 ALIGN=MIDDLE  ALT=\"x-2y=k%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k%2B4 ALIGN=MIDDLE  ALT=\"y=k%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "That looks like an intermediary form for y and so for now, we have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=k ALIGN=MIDDLE  ALT=\"z=k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k%2B4 ALIGN=MIDDLE  ALT=\"y=k%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------------------\r<BR>\n" );
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document.write( "We continue to solve for x.<BR>\n" );
document.write( "Substitute what we found for y into x-2y=k+1,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2y%2Bk%2B1 ALIGN=MIDDLE  ALT=\"x=2y%2Bk%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2%28k%2B4%29%2Bk%2B1 ALIGN=MIDDLE  ALT=\"x=2%28k%2B4%29%2Bk%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3k%2B9 ALIGN=MIDDLE  ALT=\"x=3k%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So we have this list of somewhat solved variables:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k%2B4 ALIGN=MIDDLE  ALT=\"y=k%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=k ALIGN=MIDDLE  ALT=\"z=k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------------\r<BR>\n" );
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document.write( "Now, return to the formula specified for w.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=y%2B1 ALIGN=MIDDLE  ALT=\"w=y%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, our given formula of w,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=k%2B4%2B1 ALIGN=MIDDLE  ALT=\"w=k%2B4%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, substituted for y,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=k%2B5 ALIGN=MIDDLE  ALT=\"w=k%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now our summary of nearly solved digits (variables) is this:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=k%2B5 ALIGN=MIDDLE  ALT=\"w=k%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3k%2B9 ALIGN=MIDDLE  ALT=\"x=3k%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k%2B4 ALIGN=MIDDLE  ALT=\"y=k%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=k ALIGN=MIDDLE  ALT=\"z=k\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "----------------------\r<BR>\n" );
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document.write( "SO WHAT IS k FOR FINDING EACH OF THE DIGITS?<BR>\n" );
document.write( "Our smallest digit can be at least 0 and our largest digit can be at most, 9.  <BR>\n" );
document.write( "Look at x.  Already with a constant 9 in it.  Let k=0, so the term there does not contribute anything.  The result is that:<BR>\n" );
document.write( "ANSWER FOUND__________w=5, x=9, y=4, z=0. </SPAN>\n" );
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