document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192910>Question 1192910</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let x ⧪ y = 2x/y . Suppose (a ⧪ b)⧪c − a ⧪ (b ⧪ c) = a, where a, b and c are<BR>\n" );
document.write( "nonzero integers. What is the sum of all possible positive values of c?\r<BR>\n" );
document.write( "\n" );
document.write( "Can I get help on this please? Trying to understand how to solve this? Thank you for your help and time. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824831 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=824831\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Based on the definition of the operator symbol given, we can say<BR>\n" );
document.write( "a ⧪ b = 2a/b<BR>\n" );
document.write( "b ⧪ c = 2b/c\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Furthermore,<BR>\n" );
document.write( "(a ⧪ b) ⧪ c − a ⧪ (b ⧪ c) = a<BR>\n" );
document.write( "(2a/b) ⧪ c − a ⧪ (2b/c) = a<BR>\n" );
document.write( "2(2a/b)/c − 2a/(2b/c) = a<BR>\n" );
document.write( "(4a)/(bc)− (2a/1)*(c/2b) = a<BR>\n" );
document.write( "(4a)/(bc)− (ac)/b = a<BR>\n" );
document.write( "(4a)/(bc)− (ac^2)/(bc) = a<BR>\n" );
document.write( "(4a-ac^2)/(bc) = a<BR>\n" );
document.write( "a(4-c^2)/(bc) = a<BR>\n" );
document.write( "(4-c^2)/(bc) = a/a<BR>\n" );
document.write( "(4-c^2)/(bc) = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's now multiply both sides by bc and get everything to one side.<BR>\n" );
document.write( "(4-c^2)/(bc) = 1<BR>\n" );
document.write( "4-c^2 = bc*1<BR>\n" );
document.write( "4-c^2 = bc<BR>\n" );
document.write( "0 = c^2+b*c-4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We have a quadratic where c is the variable. Treat b as a constant for now.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you were to use the quadratic formula to solve for c, then you should get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+%28-b%2Bsqrt%28b%5E2%2B16%29%29%2F2 ALIGN=MIDDLE  ALT=\"c+=+%28-b%2Bsqrt%28b%5E2%2B16%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+%28-b-sqrt%28b%5E2%2B16%29%29%2F2 ALIGN=MIDDLE  ALT=\"c+=+%28-b-sqrt%28b%5E2%2B16%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Through trial and error, and making a table of values, you should find that:<BR>\n" );
document.write( "<ul><li>b = -3 leads to c = 4 in the first equation and c = -1 in the second equation</li><li>b = 0 leads to c = 2 in the first equation and c = -2 in the second equation; however we'll ignore this case because a,b,c are nonzero</li><li>b = 3 leads to c = 1 in the first equation and c = -4 in the second equation</li></ul><BR>\n" );
document.write( "No other integer values of b will produce integer values for c\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The second equation produces nothing but negative c values, which we can ignore.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Focusing on the positive c values means:<BR>\n" );
document.write( "b = -3 leads to c = 4<BR>\n" );
document.write( "b = 3 leads to c = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add up those c values: 4+1 = 5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: 5<BR>\n" );
document.write( "</font>\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );